Teoría de la representación del grupo de Lorentz

Representation of the symmetry group of spacetime in special relativity

Hendrik Antoon Lorentz (derecha), en cuyo honor se nombró al grupo de Lorentz , y Albert Einstein , cuya teoría especial de la relatividad es la principal fuente de aplicación. Foto tomada por Paul Ehrenfest en 1921.

El grupo de Lorentz es un grupo de Lie de simetrías del espacio-tiempo de la relatividad especial . Este grupo puede realizarse como una colección de matrices , transformaciones lineales u operadores unitarios en algún espacio de Hilbert ; tiene una variedad de representaciones . ^{[nb 1]} Este grupo es significativo porque la relatividad especial junto con la mecánica cuántica son las dos teorías físicas que están más completamente establecidas, ^{[nb 2]} y la conjunción de estas dos teorías es el estudio de las representaciones unitarias de dimensión infinita del grupo de Lorentz. Estas tienen importancia histórica en la física convencional, así como conexiones con teorías actuales más especulativas.

Desarrollo

La teoría completa de las representaciones de dimensión finita del álgebra de Lie del grupo de Lorentz se deduce utilizando el marco general de la teoría de representación de álgebras de Lie semisimples . Las representaciones de dimensión finita del componente conexo del grupo de Lorentz completo O(3; 1) se obtienen empleando la correspondencia de Lie y la matriz exponencial . La teoría de representación de dimensión finita completa del grupo de recubrimiento universal (y también del grupo de espín , un recubrimiento doble) de se obtiene y se da explícitamente en términos de acción sobre un espacio de funciones en representaciones de SL ( 2 , C ) {\displaystyle {\text{SL}}(2,\mathbb {C} )} y s l ( 2 , C ) {\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )} . Los representantes de la inversión temporal y la inversión espacial se dan en inversión espacial e inversión temporal, completando la teoría de dimensión finita para el grupo de Lorentz completo. Se describen las propiedades generales de las representaciones (m, n). Se considera la acción sobre espacios de funciones, con la acción sobre armónicos esféricos y las P-funciones de Riemann apareciendo como ejemplos. El caso de dimensión infinita de representaciones unitarias irreducibles se realiza para la serie principal y la serie complementaria . Finalmente, se da la fórmula de Plancherel para , y se clasifican y realizan representaciones de SO(3, 1) para álgebras de Lie. ${\displaystyle {\text{SO}}(3;1)^{+}}$ ${\displaystyle {\text{SL}}(2,\mathbb {C} )}$ ${\displaystyle {\text{SO}}(3;1)^{+}}$ ${\displaystyle {\text{SL}}(2,\mathbb {C} )}$ ${\displaystyle {\text{SL}}(2,\mathbb {C} )}$

El desarrollo de la teoría de la representación ha seguido históricamente el desarrollo de la teoría más general de la teoría de la representación de grupos semisimples , en gran parte debido a Élie Cartan y Hermann Weyl , pero el grupo de Lorentz también ha recibido especial atención debido a su importancia en la física. Los contribuyentes notables son el físico EP Wigner y el matemático Valentine Bargmann con su programa Bargmann-Wigner , ^[1] una conclusión de la cual es, aproximadamente, que una clasificación de todas las representaciones unitarias del grupo de Lorentz no homogéneo equivale a una clasificación de todas las posibles ecuaciones de onda relativistas . ^[2] La clasificación de las representaciones irreducibles de dimensión infinita del grupo de Lorentz fue establecida por el estudiante de doctorado de Paul Dirac en física teórica, Harish-Chandra , más tarde convertido en matemático, ^{[nb 3]} en 1947. La clasificación correspondiente para fue publicada de forma independiente por Bargmann e Israel Gelfand junto con Mark Naimark en el mismo año. ${\displaystyle \mathrm {SL} (2,\mathbb {C} )}$

Aplicaciones

Muchas de las representaciones, tanto finitas como infinitas, son importantes en la física teórica. Las representaciones aparecen en la descripción de campos en la teoría clásica de campos , sobre todo el campo electromagnético , y de partículas en la mecánica cuántica relativista , así como de partículas y campos cuánticos en la teoría cuántica de campos y de varios objetos en la teoría de cuerdas y más allá. La teoría de la representación también proporciona la base teórica para el concepto de espín . La teoría entra en la relatividad general en el sentido de que en regiones suficientemente pequeñas del espacio-tiempo, la física es la de la relatividad especial. ^[3]

Las representaciones no unitarias irreducibles de dimensión finita junto con las representaciones unitarias irreducibles de dimensión infinita del grupo de Lorentz no homogéneo , el grupo de Poincaré, son las representaciones que tienen relevancia física directa. ^{[4] [5]}

Las representaciones unitarias de dimensión infinita del grupo de Lorentz aparecen por restricción de las representaciones unitarias de dimensión infinita irreducibles del grupo de Poincaré que actúan sobre los espacios de Hilbert de la mecánica cuántica relativista y la teoría cuántica de campos . Pero también son de interés matemático y de potencial relevancia física directa en otros roles que el de una mera restricción. ^[6] Hubo teorías especulativas, ^{[7] [8]} (los tensores y espinores tienen contrapartes infinitas en los expansores de Dirac y los expinores de Harish-Chandra) consistentes con la relatividad y la mecánica cuántica, pero no han encontrado ninguna aplicación física probada. Las teorías especulativas modernas potencialmente tienen ingredientes similares a los que se indican a continuación.

Teoría clásica de campos

Aunque el campo electromagnético junto con el campo gravitacional son los únicos campos clásicos que proporcionan descripciones precisas de la naturaleza, otros tipos de campos clásicos también son importantes. En el enfoque de la teoría cuántica de campos (QFT) conocido como segunda cuantificación , el punto de partida es uno o más campos clásicos, donde, por ejemplo, las funciones de onda que resuelven la ecuación de Dirac se consideran campos clásicos antes de la (segunda) cuantificación. ^[9] Aunque la segunda cuantificación y el formalismo lagrangiano asociado a ella no es un aspecto fundamental de la QFT, ^[10] es el caso de que hasta ahora todas las teorías cuánticas de campos pueden abordarse de esta manera, incluido el modelo estándar . ^[11] En estos casos, existen versiones clásicas de las ecuaciones de campo que se derivan de las ecuaciones de Euler-Lagrange derivadas del lagrangiano utilizando el principio de mínima acción . Estas ecuaciones de campo deben ser relativísticamente invariantes, y sus soluciones (que calificarán como funciones de onda relativistas según la definición siguiente) deben transformarse bajo alguna representación del grupo de Lorentz.

La acción del grupo de Lorentz sobre el espacio de configuraciones de campo (una configuración de campo es la historia espaciotemporal de una solución particular, por ejemplo, el campo electromagnético en todo el espacio durante todo el tiempo es una configuración de campo) se asemeja a la acción sobre los espacios de Hilbert de la mecánica cuántica, excepto que los corchetes del conmutador se reemplazan por corchetes de Poisson teóricos de campo . ^[9]

Mecánica cuántica relativista

Para los presentes propósitos se hace la siguiente definición: ^[12] Una función de onda relativista es un conjunto de n funciones ψ ^α en el espacio-tiempo que se transforma bajo una transformación arbitraria de Lorentz propia Λ como

$${\displaystyle \psi '^{\alpha }(x)=D{[\Lambda ]^{\alpha }}_{\beta }\psi ^{\beta }\left(\Lambda ^{-1}x\right),}$$

donde D [Λ] es una matriz n -dimensional representativa de Λ perteneciente a alguna suma directa de las ( m , n ) representaciones que se presentarán a continuación.

Las teorías de una partícula de mecánica cuántica relativista más útiles (no existen teorías de este tipo completamente consistentes) son la ecuación de Klein-Gordon ^[13] y la ecuación de Dirac ^[14] en su configuración original. Son relativísticamente invariantes y sus soluciones se transforman bajo el grupo de Lorentz como escalares de Lorentz ( ( m , n ) = (0, 0) ) y bispinores ( (0, ⁠1/2⁠ ) ​​⊕ ( ⁠1/2⁠ , 0) ) respectivamente. El campo electromagnético es una función de onda relativista según esta definición, transformándose bajo (1, 0) ⊕ (0, 1) . ^[15]

Las representaciones de dimensión infinita se pueden utilizar en el análisis de la dispersión. ^[16]

Teoría cuántica de campos

En la teoría cuántica de campos , la exigencia de invariancia relativista entra, entre otras formas, en que la matriz S necesariamente debe ser invariante de Poincaré. ^[17] Esto tiene la implicación de que hay una o más representaciones de dimensión infinita del grupo de Lorentz actuando sobre el espacio de Fock . ^{[nb 4]} Una forma de garantizar la existencia de tales representaciones es la existencia de una descripción lagrangiana (con modestos requisitos impuestos, ver la referencia) del sistema usando el formalismo canónico, de la cual se puede deducir una realización de los generadores del grupo de Lorentz. ^[18]

Las transformaciones de los operadores de campo ilustran el papel complementario desempeñado por las representaciones de dimensión finita del grupo de Lorentz y las representaciones unitarias de dimensión infinita del grupo de Poincaré, lo que da testimonio de la profunda unidad entre las matemáticas y la física. ^[19] A modo de ilustración, considere la definición de un operador de campo de n componentes : ^[20] Un operador de campo relativista es un conjunto de funciones con valores de operador n en el espacio-tiempo que se transforma bajo transformaciones de Poincaré adecuadas (Λ, a ) de acuerdo con ^{[21] [22]}

$${\displaystyle \Psi ^{\alpha }(x)\to \Psi '^{\alpha }(x)=U[\Lambda ,a]\Psi ^{\alpha }(x)U^{-1}\left[\Lambda ,a\right]=D{\left[\Lambda ^{-1}\right]^{\alpha }}_{\beta }\Psi ^{\beta }(\Lambda x+a)}$$

Aquí U [Λ, a] es el operador unitario que representa (Λ, a) en el espacio de Hilbert en el que está definido Ψ y D es una representación n -dimensional del grupo de Lorentz. La regla de transformación es el segundo axioma de Wightman de la teoría cuántica de campos.

Por consideraciones de restricciones diferenciales a las que debe estar sujeto el operador de campo para describir una única partícula con masa definida m y espín s (o helicidad), se deduce que ^{[23] [nb 5]}

donde a ^† , a se interpretan como operadores de creación y aniquilación respectivamente. El operador de creación a ^† se transforma según ^{[23] [24]}

$${\displaystyle a^{\dagger }(\mathbf {p} ,\sigma )\rightarrow a'^{\dagger }\left(\mathbf {p} ,\sigma \right)=U[\Lambda ]a^{\dagger }(\mathbf {p} ,\sigma )U\left[\Lambda ^{-1}\right]=a^{\dagger }(\Lambda \mathbf {p} ,\rho )D^{(s)}{\left[R(\Lambda ,\mathbf {p} )^{-1}\right]^{\rho }}_{\sigma },}$$

y de manera similar para el operador de aniquilación. El punto a destacar es que el operador de campo se transforma de acuerdo con una representación no unitaria de dimensión finita del grupo de Lorentz, mientras que el operador de creación se transforma bajo la representación unitaria de dimensión infinita del grupo de Poincaré caracterizado por la masa y el espín ( m , s ) de la partícula. La conexión entre los dos son las funciones de onda , también llamadas funciones de coeficientes.

$${\displaystyle u^{\alpha }(\mathbf {p} ,\sigma )e^{ip\cdot x},\quad v^{\alpha }(\mathbf {p} ,\sigma )e^{-ip\cdot x}}$$

que llevan tanto los índices ( x , α ) operados por transformaciones de Lorentz como los índices ( p , σ ) operados por transformaciones de Poincaré. Esto puede llamarse la conexión de Lorentz-Poincaré. ^[25] Para mostrar la conexión, someta ambos lados de la ecuación (X1) a una transformación de Lorentz que resulte en, por ejemplo , u ,

$${\displaystyle {D[\Lambda ]^{\alpha }}_{\alpha '}u^{\alpha '}(\mathbf {p} ,\lambda )={D^{(s)}[R(\Lambda ,\mathbf {p} )]^{\lambda '}}_{\lambda }u^{\alpha }\left(\Lambda \mathbf {p} ,\lambda '\right),}$$

donde D es el grupo de Lorentz no unitario representante de Λ y D ^{( s )} es un representante unitario de la llamada rotación de Wigner R asociada a Λ y p que deriva de la representación del grupo de Poincaré, y s es el espín de la partícula.

Todas las fórmulas anteriores, incluyendo la definición del operador de campo en términos de operadores de creación y aniquilación, así como las ecuaciones diferenciales satisfechas por el operador de campo para una partícula con masa especificada, espín y la representación ( m , n ) bajo la cual se supone que se transforma, ^{[nb 6]} y también la de la función de onda, se pueden derivar únicamente de consideraciones teóricas de grupo una vez que se dan los marcos de la mecánica cuántica y la relatividad especial. ^{[nb 7]}

Teorías especulativas

En las teorías en las que el espacio-tiempo puede tener más de D = 4 dimensiones, los grupos de Lorentz generalizados O( D − 1; 1) de la dimensión apropiada toman el lugar de O(3; 1) . ^{[nb 8]}

El requisito de la invariancia de Lorentz adquiere quizás su efecto más dramático en la teoría de cuerdas . Las cuerdas relativistas clásicas pueden manejarse en el marco lagrangiano utilizando la acción de Nambu-Goto . ^[26] Esto da como resultado una teoría relativista invariante en cualquier dimensión del espacio-tiempo. ^[27] Pero resulta que la teoría de cuerdas bosónicas abiertas y cerradas (la teoría de cuerdas más simple) es imposible de cuantificar de tal manera que el grupo de Lorentz esté representado en el espacio de estados (un espacio de Hilbert ) a menos que la dimensión del espacio-tiempo sea 26. ^[28] El resultado correspondiente para la teoría de supercuerdas se deduce nuevamente exigiendo la invariancia de Lorentz, pero ahora con supersimetría . En estas teorías, el álgebra de Poincaré se reemplaza por un álgebra de supersimetría que es un álgebra de Lie graduada en Z 2 que extiende el álgebra de Poincaré. La estructura de un álgebra de este tipo está determinada en gran medida por las exigencias de la invariancia de Lorentz. En particular, los operadores fermiónicos (grado 1 ) pertenecen a un (0, ⁠1/2⁠ ) ​​o ( ⁠1/2⁠ , 0) espacio de representación del álgebra de Lie de Lorentz (ordinaria). ^[29] La única dimensión posible del espacio-tiempo en tales teorías es 10. ^[30]

Representaciones de dimensión finita

La teoría de la representación de grupos en general, y de grupos de Lie en particular, es un tema muy rico. El grupo de Lorentz tiene algunas propiedades que lo hacen "agradable" y otras que lo hacen "poco agradable" dentro del contexto de la teoría de la representación; el grupo es simple y por lo tanto semisimple , pero no es conexo , y ninguno de sus componentes es simplemente conexo . Además, el grupo de Lorentz no es compacto . ^[31]

Para las representaciones de dimensión finita, la presencia de semisimplicidad significa que el grupo de Lorentz puede ser tratado de la misma manera que otros grupos semisimples usando una teoría bien desarrollada. Además, todas las representaciones se construyen a partir de las irreducibles , ya que el álgebra de Lie posee la propiedad de reducibilidad completa . ^{[nb 9] [32]} Pero, la no compacidad del grupo de Lorentz, en combinación con la falta de conectividad simple, no puede ser tratada en todos los aspectos como en el marco simple que se aplica a los grupos compactos simplemente conectados. La no compacidad implica, para un grupo de Lie simple conectado, que no existen representaciones unitarias de dimensión finita no triviales. ^[33] La falta de conectividad simple da lugar a representaciones de espín del grupo. ^[34] La no conectividad significa que, para las representaciones del grupo de Lorentz completo, la inversión del tiempo y la inversión de la orientación espacial deben tratarse por separado. ^{[35] [36]}

Historia

El desarrollo de la teoría de representación de dimensión finita del grupo de Lorentz sigue en su mayor parte al de la teoría de representación en general. La teoría de Lie se originó con Sophus Lie en 1873. ^{[37] [38]} En 1888, la clasificación de álgebras de Lie simples fue esencialmente completada por Wilhelm Killing . ^{[39] [40]} En 1913, el teorema de mayor peso para representaciones de álgebras de Lie simples, el camino que se seguirá aquí, fue completado por Élie Cartan . ^{[41] [42]} Richard Brauer fue durante el período de 1935-38 en gran parte responsable del desarrollo de las matrices de Weyl-Brauer que describen cómo las representaciones de espín del álgebra de Lie de Lorentz pueden integrarse en las álgebras de Clifford . ^{[43] [44]} El grupo de Lorentz también ha recibido históricamente atención especial en la teoría de la representación, consulte Historia de las representaciones unitarias de dimensión infinita a continuación, debido a su importancia excepcional en la física. Los matemáticos Hermann Weyl ^{[41] [45] [37] [46] [47]} y Harish-Chandra ^{[48] [49]} y los físicos Eugene Wigner ^{[50] [51]} y Valentine Bargmann ^{[52] [53] [54]} hicieron contribuciones sustanciales tanto a la teoría de la representación general como en particular al grupo de Lorentz. ^[55] El físico Paul Dirac fue quizás el primero en unir manifiestamente todo en una aplicación práctica de gran importancia duradera con la ecuación de Dirac en 1928. ^{[56] [57] [nb 10]}

El álgebra de Lie

Wilhelm Killing , descubridor independiente de las álgebras de Lie . Las álgebras de Lie simples fueron clasificadas por primera vez por él en 1888.

Esta sección aborda las representaciones lineales complejas irreducibles de la complejización del álgebra de Lie del grupo de Lorentz. Una base conveniente para está dada por los tres generadores J _i de rotaciones y los tres generadores K _i de impulsos . Se dan explícitamente en las convenciones y bases del álgebra de Lie. ${\displaystyle {\mathfrak {so}}(3;1)_{\mathbb {C} }}$ ${\displaystyle {\mathfrak {so}}(3;1)}$ ${\displaystyle {\mathfrak {so}}(3;1)}$

El álgebra de Lie se complejiza y la base se cambia a los componentes de sus dos ideales ^[58] $${\displaystyle \mathbf {A} ={\frac {\mathbf {J} +i\mathbf {K} }{2}},\quad \mathbf {B} ={\frac {\mathbf {J} -i\mathbf {K} }{2}}.}$$

Las componentes de A = ( A ₁ ,  A ₂ ,  A ₃ ) y B = ( B ₁ ,  B ₂ ,  B ₃ ) satisfacen por separado las relaciones de conmutación del álgebra de Lie y, además, conmutan entre sí, ^[59] ${\displaystyle {\mathfrak {su}}(2)}$

$${\displaystyle \left[A_{i},A_{j}\right]=i\varepsilon _{ijk}A_{k},\quad \left[B_{i},B_{j}\right]=i\varepsilon _{ijk}B_{k},\quad \left[A_{i},B_{j}\right]=0,}$$

donde i ,  j ,  k son índices que toman cada uno los valores 1, 2, 3 y ε _ijk es el símbolo tridimensional de Levi-Civita . Sea y el intervalo lineal complejo de A y B respectivamente. ${\displaystyle \mathbf {A} _{\mathbb {C} }}$ ${\displaystyle \mathbf {B} _{\mathbb {C} }}$

Se tienen los isomorfismos ^{[60] [nb 11]}

¿Dónde está la complejización de ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )}$ ${\displaystyle {\mathfrak {su}}(2)\cong \mathbf {A} \cong \mathbf {B} .}$

La utilidad de estos isomorfismos proviene del hecho de que se conocen todas las representaciones irreducibles de ${\displaystyle {\mathfrak {su}}(2)}$ , y por lo tanto todas las representaciones lineales complejas irreducibles de . La representación lineal compleja irreducible de es isomorfa a una de las representaciones de mayor peso . Estas se dan explícitamente en representaciones lineales complejas de s l ( 2 , C ) . {\displaystyle {\mathfrak {sl}}(2,\mathbb {C} ).} ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} ),}$ ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )}$

El truco unitario

Hermann Weyl , inventor del truco unitario . Existen varios conceptos y fórmulas en la teoría de la representación que llevan el nombre de Weyl, por ejemplo, el grupo de Weyl y la fórmula de caracteres de Weyl .

El álgebra de Lie es el álgebra de Lie de Contiene el subgrupo compacto SU(2) × SU(2) con álgebra de Lie Esta última es una forma real compacta de Por lo tanto, a partir del primer enunciado del truco unitario, las representaciones de SU(2) × SU(2) están en correspondencia biunívoca con las representaciones holomorfas de ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )\oplus {\mathfrak {sl}}(2,\mathbb {C} )}$ ${\displaystyle {\text{SL}}(2,\mathbb {C} )\times {\text{SL}}(2,\mathbb {C} ).}$ ${\displaystyle {\mathfrak {su}}(2)\oplus {\mathfrak {su}}(2).}$ ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )\oplus {\mathfrak {sl}}(2,\mathbb {C} ).}$ ${\displaystyle {\text{SL}}(2,\mathbb {C} )\times {\text{SL}}(2,\mathbb {C} ).}$

Por compacidad, el teorema de Peter-Weyl se aplica a SU(2) × SU(2) , ^[61] y, por lo tanto, se puede apelar a la ortonormalidad de los caracteres irreducibles . Las representaciones unitarias irreducibles de SU(2) × SU(2) son precisamente los productos tensoriales de las representaciones unitarias irreducibles de SU(2) . ^[62]

Apelando a la simple conexión, se aplica el segundo enunciado del truco unitario. Los objetos de la siguiente lista se corresponden uno a uno:

• Representaciones holomorfas de ${\displaystyle {\text{SL}}(2,\mathbb {C} )\times {\text{SL}}(2,\mathbb {C} )}$
• Representaciones suaves de SU(2) × SU(2)
• Representaciones lineales reales de ${\displaystyle {\mathfrak {su}}(2)\oplus {\mathfrak {su}}(2)}$
• Representaciones lineales complejas de ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )\oplus {\mathfrak {sl}}(2,\mathbb {C} )}$

Los productos tensoriales de representaciones aparecen en el nivel del álgebra de Lie como cualquiera de los siguientes: ^{[nb 12]}

donde Id es el operador identidad. Aquí se pretende la última interpretación, que se desprende de (G6) . Las representaciones de mayor peso de están indexadas por μ para μ = 0, 1/2, 1, ... (Los pesos más altos son en realidad 2 μ = 0, 1, 2, ... , pero la notación aquí está adaptada a la de ) Los productos tensoriales de dos de estos factores lineales complejos forman entonces las representaciones lineales complejas irreducibles de ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )}$ ${\displaystyle {\mathfrak {so}}(3;1).}$ ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )\oplus {\mathfrak {sl}}(2,\mathbb {C} ).}$

Finalmente, las representaciones -lineales de las formas reales de la extrema izquierda, , y de la extrema derecha, ^{[nb 13]} en (A1) se obtienen a partir de las representaciones -lineales de caracterizadas en el párrafo anterior. ${\displaystyle \mathbb {R} }$ ${\displaystyle {\mathfrak {so}}(3;1)}$ ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} ),}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )\oplus {\mathfrak {sl}}(2,\mathbb {C} )}$

El (micras,no)-representaciones de sl(2, C)

Las representaciones lineales complejas de la complejización de obtenidas mediante isomorfismos en (A1) , se corresponden biunívocamente con las representaciones lineales reales de ^[63] El conjunto de todas las representaciones lineales irreducibles reales de están, por tanto, indexadas por un par ( μ , ν ) . Las lineales complejas, que corresponden precisamente a la complejización de las representaciones lineales reales, son de la forma ( μ , 0) , mientras que las lineales conjugadas son (0, ν ) . ^[63] Todas las demás son solo lineales reales. Las propiedades de linealidad se derivan de la inyección canónica, la extrema derecha en (A1) , de en su complejización. Las representaciones en la forma ( ν , ν ) o ( μ , ν ) ⊕ ( ν , μ ) están dadas por matrices reales (estas últimas no son irreducibles). Explícitamente, las representaciones lineales reales ( μ , ν ) de son donde son las representaciones irreducibles lineales complejas de y sus representaciones conjugadas complejas. (El etiquetado suele ser en la literatura matemática 0, 1, 2, ... , pero aquí se eligen semienteros para cumplir con el etiquetado del álgebra de Lie). Aquí el producto tensorial se interpreta en el sentido anterior de (A0) . Estas representaciones se realizan concretamente a continuación. ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} ),{\mathfrak {sl}}(2,\mathbb {C} )_{\mathbb {C} },}$ ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} ).}$ ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )}$ ${\displaystyle {\mathfrak {su}}(2)}$ ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )}$ ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )}$ $${\displaystyle \varphi _{\mu ,\nu }(X)=\left(\varphi _{\mu }\otimes {\overline {\varphi _{\nu }}}\right)(X)=\varphi _{\mu }(X)\otimes \operatorname {Id} _{\nu +1}+\operatorname {Id} _{\mu +1}\otimes {\overline {\varphi _{\nu }(X)}},\qquad X\in {\mathfrak {sl}}(2,\mathbb {C} )}$$ ${\textstyle \varphi _{\mu },\mu =0,{\tfrac {1}{2}},1,{\tfrac {3}{2}},\ldots }$ ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )}$ ${\displaystyle {\overline {\varphi _{\nu }}},\nu =0,{\tfrac {1}{2}},1,{\tfrac {3}{2}},\ldots }$ ${\displaystyle {\mathfrak {so}}(3,1)}$

El (metro,norte)-representaciones de so(3; 1)

A través de los isomorfismos mostrados en (A1) y el conocimiento de las representaciones irreducibles lineales complejas de al resolver para J y K , se obtienen todas las representaciones irreducibles de y, por restricción, las de . Las representaciones de obtenidas de esta manera son lineales reales (y no lineales complejas o conjugadas) porque el álgebra no está cerrada tras la conjugación, pero siguen siendo irreducibles. ^[60] Dado que es semisimple , ^[60] todas sus representaciones pueden construirse como sumas directas de las irreducibles. ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )\oplus {\mathfrak {sl}}(2,\mathbb {C} )}$ ${\displaystyle {\mathfrak {so}}(3;1)_{\mathbb {C} },}$ ${\displaystyle {\mathfrak {so}}(3;1)}$ ${\displaystyle {\mathfrak {so}}(3;1)}$ ${\displaystyle {\mathfrak {so}}(3;1)}$

Así, las representaciones irreducibles de dimensión finita del álgebra de Lorentz se clasifican mediante un par ordenado de semienteros m = μ y n = ν , convencionalmente escritos como uno de donde V es un espacio vectorial de dimensión finita. Estos están dados de manera única, hasta una transformación de similitud , por ^{[nb 14]} $${\displaystyle (m,n)\equiv \pi _{(m,n)}:{\mathfrak {so}}(3;1)\to {\mathfrak {gl}}(V),}$$

donde 1 _n es la matriz unitaria de dimensión n y son las representaciones irreducibles de dimensión (2 n  + 1) de las también denominadas matrices de espín o matrices de momento angular . Estas se dan explícitamente como ^[64] donde δ denota el delta de Kronecker . En componentes, con − m ≤ a , a′ ≤ m , − n ≤ b , b′ ≤ n , las representaciones se dan por ^[65] $${\displaystyle \mathbf {J} ^{(n)}=\left(J_{1}^{(n)},J_{2}^{(n)},J_{3}^{(n)}\right)}$$ ${\displaystyle {\mathfrak {so}}(3)\cong {\mathfrak {su}}(2)}$ $${\displaystyle {\begin{aligned}\left(J_{1}^{(j)}\right)_{a'a}&={\frac {1}{2}}\left({\sqrt {(j-a)(j+a+1)}}\delta _{a',a+1}+{\sqrt {(j+a)(j-a+1)}}\delta _{a',a-1}\right)\\\left(J_{2}^{(j)}\right)_{a'a}&={\frac {1}{2i}}\left({\sqrt {(j-a)(j+a+1)}}\delta _{a',a+1}-{\sqrt {(j+a)(j-a+1)}}\delta _{a',a-1}\right)\\\left(J_{3}^{(j)}\right)_{a'a}&=a\delta _{a',a}\end{aligned}}}$$ $${\displaystyle {\begin{aligned}\left(\pi _{(m,n)}\left(J_{i}\right)\right)_{a'b',ab}&=\delta _{b'b}\left(J_{i}^{(m)}\right)_{a'a}+\delta _{a'a}\left(J_{i}^{(n)}\right)_{b'b}\\\left(\pi _{(m,n)}\left(K_{i}\right)\right)_{a'b',ab}&=-i\left(\delta _{b'b}\left(J_{i}^{(m)}\right)_{a'a}-\delta _{a'a}\left(J_{i}^{(n)}\right)_{b'b}\right)\end{aligned}}}$$

Representaciones comunes

• La representación (0, 0) es la representación trivial unidimensional y es llevada a cabo por teorías de campos escalares relativistas .
• Los generadores de supersimetría fermiónica se transforman bajo uno de los (0,  ⁠1/2⁠ ) ​​o ( ⁠1/2⁠ , 0) representaciones (espinores de Weyl). ^[29]
• El cuadrimpulso de una partícula (ya sea sin masa o con masa ) se transforma bajo la ( ⁠1/2⁠ ,  ⁠1/2) representación , un cuatro-vector.
• Un ejemplo físico de un campo tensorial simétrico sin traza (1,1) es la parte sin traza ^{[nb 15]} del tensor de energía-momento T ^μν . ^{[66] [nb 16]}

Sumas directas fuera de la diagonal

Dado que para cualquier representación irreducible para la cual m ≠ n es esencial operar sobre el cuerpo de los números complejos , la suma directa de las representaciones ( m ,  n ) y ( n ,  m ) tiene particular relevancia para la física, ya que permite utilizar operadores lineales sobre números reales .

• ( ⁠1/2⁠ , 0) ⊕ (0,  ⁠1/2) es la representación del bispinor . Véase también espinor de Dirac y espinores y bispinores de Weyl a continuación .
• (1,  ⁠1/2⁠ ) ​​⊕ ( ⁠1/2⁠ , 1) es la representación del campo Rarita-Schwinger .
• ( ⁠3/2⁠ , 0) ⊕ (0, ⁠3/2) sería la simetría del gravitino hipotético . [ nb ^17] Se puede obtener de (1,  ⁠1/2⁠ ) ​​⊕ ( ⁠1/2⁠ , 1) representación.
• (1, 0) ⊕ (0, 1) es la representación de un campo de 2 formas invariante de paridad (también conocido como forma de curvatura ). El tensor del campo electromagnético se transforma bajo esta representación.

El grupo

El enfoque de esta sección se basa en teoremas que, a su vez, se basan en la correspondencia de Lie fundamental . ^[67] La ​​correspondencia de Lie es en esencia un diccionario entre grupos de Lie conectados y álgebras de Lie. ^[68] El vínculo entre ellos es la aplicación exponencial del álgebra de Lie al grupo de Lie, denotada ${\displaystyle \exp :{\mathfrak {g}}\to G.}$

Si para algún espacio vectorial V es una representación, una representación Π del componente conexo de G se define por ${\displaystyle \pi :{\mathfrak {g}}\to {\mathfrak {gl}}(V)}$

Esta definición se aplica independientemente de que la representación resultante sea proyectiva o no.

Sobreyectividad del mapa exponencial para SO(3, 1)

Desde un punto de vista práctico, es importante si la primera fórmula en (G2) se puede utilizar para todos los elementos del grupo . Se cumple para todos los , sin embargo, en el caso general, por ejemplo para , no todos los g ∈ G están en la imagen de exp . ${\displaystyle X\in {\mathfrak {g}}}$ ${\displaystyle {\text{SL}}(2,\mathbb {C} )}$

Pero es sobreyectiva. Una forma de mostrar esto es hacer uso del isomorfismo, siendo este último el grupo de Möbius . Es un cociente de (ver el artículo vinculado). La función cociente se denota con La función es sobreyectiva. ^[69] Aplicar (Lie) con π siendo la diferencial de p en la identidad. Entonces ${\displaystyle \exp :{\mathfrak {so}}(3;1)\to {\text{SO}}(3;1)^{+}}$ ${\displaystyle {\text{SO}}(3;1)^{+}\cong {\text{PGL}}(2,\mathbb {C} ),}$ ${\displaystyle {\text{GL}}(n,\mathbb {C} )}$ ${\displaystyle p:{\text{GL}}(n,\mathbb {C} )\to {\text{PGL}}(2,\mathbb {C} ).}$ ${\displaystyle \exp :{\mathfrak {gl}}(n,\mathbb {C} )\to {\text{GL}}(n,\mathbb {C} )}$

$${\displaystyle \forall X\in {\mathfrak {gl}}(n,\mathbb {C} ):\quad p(\exp(iX))=\exp(i\pi (X)).}$$

Como el lado izquierdo es sobreyectivo (tanto exp como p lo son), el lado derecho es sobreyectivo y, por lo tanto, es sobreyectivo. ^[70] Finalmente, recicle el argumento una vez más, pero ahora con el isomorfismo conocido entre SO(3; 1) ⁺ y para encontrar que exp es sobreyectivo para el componente conexo del grupo de Lorentz. ${\displaystyle \exp :{\mathfrak {pgl}}(2,\mathbb {C} )\to {\text{PGL}}(2,\mathbb {C} )}$ ${\displaystyle {\text{PGL}}(2,\mathbb {C} )}$

Grupo fundamental

El grupo de Lorentz está doblemente conexo , es decir, π ₁ (SO(3; 1)) es un grupo con dos clases de equivalencia de bucles como sus elementos.

Prueba

Para representar el grupo fundamental de SO(3; 1) ⁺ , se considera la topología de su grupo de recubrimiento SL ( 2 , C ) {\displaystyle {\text{SL}}(2,\mathbb {C} )}. Por el teorema de descomposición polar , cualquier matriz puede expresarse de forma única como ^[71] ${\displaystyle \lambda \in {\text{SL}}(2,\mathbb {C} )}$

$${\displaystyle \lambda =ue^{h},}$$

donde u es unitaria con determinante uno, por lo tanto en SU(2) , y h es hermítica con traza cero. Las condiciones de traza y determinante implican: ^[72] $${\displaystyle {\begin{aligned}h&={\begin{pmatrix}c&a-ib\\a+ib&-c\end{pmatrix}}&&(a,b,c)\in \mathbb {R} ^{3}\\[4pt]u&={\begin{pmatrix}d+ie&f+ig\\-f+ig&d-ie\end{pmatrix}}&&(d,e,f,g)\in \mathbb {R} ^{4}{\text{ subject to }}d^{2}+e^{2}+f^{2}+g^{2}=1.\end{aligned}}}$$

La función biunívoca manifiestamente continua es un homeomorfismo con inverso continuo dado por (el lugar geométrico de u se identifica con ) ${\displaystyle \mathbb {S} ^{3}\subset \mathbb {R} ^{4}}$

$${\displaystyle {\begin{cases}\mathbb {R} ^{3}\times \mathbb {S} ^{3}\to {\text{SL}}(2,\mathbb {C} )\\(r,s)\mapsto u(s)e^{h(r)}\end{cases}}}$$

mostrando explícitamente que está simplemente conectado. Pero, ¿dónde está el centro de ? Identificar λ y − λ equivale a identificar u con − u , lo que a su vez equivale a identificar puntos antípodas en Por lo tanto, topológicamente, ^[72] ${\displaystyle {\text{SL}}(2,\mathbb {C} )}$ ${\displaystyle {\text{SO}}(3;1)\cong {\text{SL}}(2,\mathbb {C} )/\{\pm I\},}$ ${\displaystyle \{\pm I\}}$ ${\displaystyle {\text{SL}}(2,\mathbb {C} )}$ ${\displaystyle \mathbb {S} ^{3}.}$ $${\displaystyle {\text{SO}}(3;1)\cong \mathbb {R} ^{3}\times (\mathbb {S} ^{3}/\mathbb {Z} _{2}),}$$

donde el último factor no está simplemente conexo: Geométricamente, se ve (para fines de visualización, puede reemplazarse por ) que un camino desde u hasta − u en es un bucle en ya que u y − u son puntos antípodas, y que no es contráctil a un punto. Pero un camino desde u hasta − u , de allí a u nuevamente, un bucle en y un doble bucle (considerando p ( ue ^h ) = p (− ue ^h ) , donde es la función de recubrimiento) en que es contráctil a un punto (alejándose continuamente de − u "arriba" en y encogiendo el camino allí hasta el punto u ). ^[72] Por lo tanto, π ₁ (SO(3; 1)) es un grupo con dos clases de equivalencia de bucles como sus elementos, o dicho de manera más simple, SO(3; 1) está doblemente conexo . ${\displaystyle \mathbb {S} ^{3}}$ ${\displaystyle \mathbb {S} ^{2}}$ ${\displaystyle SU(2)\cong \mathbb {S} ^{3}}$ ${\displaystyle \mathbb {S} ^{3}/\mathbb {Z} _{2}}$ ${\displaystyle \mathbb {S} ^{3}}$ ${\displaystyle p:{\text{SL}}(2,\mathbb {C} )\to {\text{SO}}(3;1)}$ ${\displaystyle \mathbb {S} ^{3}/\mathbb {Z} _{2}}$ ${\displaystyle \mathbb {S} ^{3}}$

Representaciones proyectivas

Como π ₁ (SO(3; 1) ⁺ ) tiene dos elementos, algunas representaciones del álgebra de Lie producirán representaciones proyectivas . ^{[73] [nb 18]} Una vez que se sabe si una representación es proyectiva, la fórmula (G2) se aplica a todos los elementos del grupo y a todas las representaciones, incluidas las proyectivas, con el entendimiento de que el representante de un elemento del grupo dependerá de qué elemento del álgebra de Lie (la X en (G2) ) se utiliza para representar el elemento del grupo en la representación estándar.

Para el grupo de Lorentz, la representación ( m , n ) es proyectiva cuando m + n es un semientero. Véase § Espinores.

Para una representación proyectiva Π de SO(3; 1) ⁺ , se cumple que ^[72]

dado que cualquier bucle en SO(3; 1) ⁺ atravesado dos veces, debido a la doble conectividad, es contráctil a un punto, de modo que su clase de homotopía es la de una función constante. De ello se deduce que Π es una función de doble valor. No es posible elegir consistentemente un signo para obtener una representación continua de todo SO(3; 1) ⁺ , pero esto es posible localmente alrededor de cualquier punto. ^[33]

El grupo de recubrimiento SL(2, C)

Consideremos como un álgebra de Lie real con base ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )}$

$${\displaystyle \left({\frac {1}{2}}\sigma _{1},{\frac {1}{2}}\sigma _{2},{\frac {1}{2}}\sigma _{3},{\frac {i}{2}}\sigma _{1},{\frac {i}{2}}\sigma _{2},{\frac {i}{2}}\sigma _{3}\right)\equiv (j_{1},j_{2},j_{3},k_{1},k_{2},k_{3}),}$$

donde las sigmas son las matrices de Pauli . De las relaciones

se obtiene

que son exactamente la forma de la versión tridimensional de las relaciones de conmutación para (ver las convenciones y bases del álgebra de Lie a continuación). Por lo tanto, la función J _i ↔ j _i , K _i ↔ k _i , extendida por linealidad es un isomorfismo. Como es simplemente conexo, es el grupo de recubrimiento universal de SO(3; 1) ⁺ . ${\displaystyle {\mathfrak {so}}(3;1)}$ ${\displaystyle {\text{SL}}(2,\mathbb {C} )}$

Más sobre los grupos de cobertura en general y la cobertura del grupo de Lorentz en particular ${\displaystyle {\text{SL}}(2,\mathbb {C} )}$

Una visión geométrica

EP Wigner investigó el grupo de Lorentz en profundidad y es conocido por las ecuaciones de Bargmann-Wigner . La realización del grupo de recubrimiento que se da aquí proviene de su artículo de 1939.

Sea p _g ( t ), 0 ≤ t ≤ 1 un camino desde 1 ∈ SO(3; 1) ⁺ hasta g ∈ SO(3; 1) ⁺ , denotemos su clase de homotopía por [ p _g ] y sea π _g el conjunto de todas esas clases de homotopía. Definamos el conjunto

y dotarlo de la operación de multiplicación

¿Dónde está la multiplicación de la ruta de y : ${\displaystyle p_{12}}$ ${\displaystyle p_{1}}$ ${\displaystyle p_{2}}$

$${\displaystyle p_{12}(t)=(p_{1}*p_{2})(t)={\begin{cases}p_{1}(2t)&0\leqslant t\leqslant {\tfrac {1}{2}}\\p_{2}(2t-1)&{\tfrac {1}{2}}\leqslant t\leqslant 1\end{cases}}}$$

Con esta multiplicación, G se convierte en un grupo isomorfo al ^[74] grupo de recubrimiento universal de SO(3; 1) ⁺ . Dado que cada π _g tiene dos elementos, por la construcción anterior, hay una función de recubrimiento 2:1 p  : G → SO(3; 1) ⁺ . Según la teoría de grupos de recubrimiento , las álgebras de Lie y de G son todas isomorfas. La función de recubrimiento p  : G → SO(3; 1) ⁺ está dada simplemente por p ( g , [ p _g ]) = g . ${\displaystyle {\text{SL}}(2,\mathbb {C} ),}$ ${\displaystyle {\mathfrak {so}}(3;1),{\mathfrak {sl}}(2,\mathbb {C} )}$ ${\displaystyle {\mathfrak {g}}}$

Una visión algebraica

Para una visión algebraica del grupo de recubrimiento universal, sea que actúe sobre el conjunto de todas las matrices hermíticas 2 × 2 mediante la operación ^[72] ${\displaystyle {\text{SL}}(2,\mathbb {C} )}$ ${\displaystyle {\mathfrak {h}}}$

La acción sobre es lineal. Un elemento de puede escribirse en la forma ${\displaystyle {\mathfrak {h}}}$ ${\displaystyle {\mathfrak {h}}}$

La función P es un homomorfismo de grupo en Por lo tanto, es una representación de 4 dimensiones de . Su núcleo debe, en particular, tomar la matriz identidad para sí misma, A ^† IA = A ^† A = I y, por lo tanto, A ^† = A ⁻¹ . Por lo tanto, AX = XA para A en el núcleo, por lo que, por el lema de Schur , ^{[nb 19]} A es un múltiplo de la identidad, que debe ser ± I ya que det A = 1 . ^[75] El espacio se asigna al espacio de Minkowski M ⁴ , mediante ${\displaystyle {\text{GL}}({\mathfrak {h}})\subset {\text{End}}({\mathfrak {h}}).}$ ${\displaystyle \mathbf {P} :{\text{SL}}(2,\mathbb {C} )\to {\text{GL}}({\mathfrak {h}})}$ ${\displaystyle {\text{SL}}(2,\mathbb {C} )}$ ${\displaystyle {\mathfrak {h}}}$

La acción de P ( A ) sobre preserva los determinantes. La representación inducida p de sobre a través del isomorfismo anterior, dada por ${\displaystyle {\mathfrak {h}}}$ ${\displaystyle {\text{SL}}(2,\mathbb {C} )}$ ${\displaystyle \mathbb {R} ^{4},}$

conserva el producto interno de Lorentz ya que $${\displaystyle -\det X=\xi _{1}^{2}+\xi _{2}^{2}+\xi _{3}^{2}-\xi _{4}^{2}=x^{2}+y^{2}+z^{2}-t^{2}.}$$

Esto significa que p ( A ) pertenece al grupo completo de Lorentz SO(3; 1) . Por el teorema principal de conexidad , dado que es conexo, su imagen bajo p en SO(3; 1) es conexa y, por lo tanto, está contenida en SO(3; 1) ⁺ . ${\displaystyle {\text{SL}}(2,\mathbb {C} )}$

Se puede demostrar que el mapa de Lie de es un isomorfismo del álgebra de Lie: ^{[nb 20]} El mapa P también es sobreyectivo. ^{[nb 21]} ${\displaystyle \mathbf {p} :{\text{SL}}(2,\mathbb {C} )\to {\text{SO}}(3;1)^{+},}$ ${\displaystyle \pi :{\mathfrak {sl}}(2,\mathbb {C} )\to {\mathfrak {so}}(3;1).}$

Por lo tanto , puesto que está simplemente conexo, es el grupo de recubrimiento universal de SO(3; 1) ⁺ , isomorfo al grupo G de arriba. ${\displaystyle {\text{SL}}(2,\mathbb {C} )}$

No sobreyectividad de la función exponencial para SL(2, C)

Este diagrama muestra la red de aplicaciones analizadas en el texto. Aquí V es un espacio vectorial de dimensión finita que contiene representaciones de y es la función exponencial, p es la función que cubre sobre SO(3; 1) ⁺ y σ es el isomorfismo del álgebra de Lie inducido por ella. Las funciones Π, π y las dos Φ son representaciones. La imagen es sólo parcialmente verdadera cuando Π es proyectiva. ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} ),{\mathfrak {so}}(3;1),}$ ${\displaystyle {\text{SL}}(2,\mathbb {C} )}$ ${\displaystyle {\text{SO}}(3;1)^{+}.}$ ${\displaystyle \exp }$ ${\displaystyle {\text{SL}}(2,\mathbb {C} ),}$

La función exponencial no es sobreyectiva. ^[76] La matriz ${\displaystyle \exp :{\mathfrak {sl}}(2,\mathbb {C} )\to {\text{SL}}(2,\mathbb {C} )}$

está dentro pero no hay tal que q = exp( Q ) . ^{[nb 22]} ${\displaystyle {\text{SL}}(2,\mathbb {C} ),}$ ${\displaystyle Q\in {\mathfrak {sl}}(2,\mathbb {C} )}$

En general, si g es un elemento de un grupo de Lie conexo G con álgebra de Lie entonces, por (Lie) , ${\displaystyle {\mathfrak {g}},}$

La matriz q se puede escribir

Realización de representaciones deSL(2, C)ysl(2, C)y sus álgebras de Lie

Las representaciones lineales complejas de y son más sencillas de obtener que las representaciones. Pueden escribirse (y normalmente se escriben) desde cero. Las representaciones de grupo holomorfas (lo que significa que la representación del álgebra de Lie correspondiente es lineal compleja) están relacionadas con las representaciones del álgebra de Lie lineal compleja por exponenciación. Las representaciones lineales reales de son exactamente las ( μ , ν ) -representaciones. También pueden ser exponenciadas. Las ( μ , 0) -representaciones son lineales complejas y son (isomorfas a) las representaciones de mayor peso. Estas suelen estar indexadas con un solo entero (pero aquí se utilizan medios enteros). ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )}$ ${\displaystyle {\text{SL}}(2,\mathbb {C} )}$ ${\displaystyle {\mathfrak {so}}(3;1)^{+}}$ ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )}$

En esta sección se utiliza la convención matemática por conveniencia. Los elementos del álgebra de Lie difieren en un factor de i y no hay ningún factor de i en la función exponencial en comparación con la convención de física utilizada en otros lugares. Sea la base de [ ^77] ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )}$

Esta elección de base y de notación es estándar en la literatura matemática.

Representaciones lineales complejas

Las representaciones holomórficas irreducibles de dimensión ( n +1) se pueden realizar en el espacio de polinomios homogéneos de grado n en 2 variables ^{[78] [79]} cuyos elementos son ${\displaystyle {\text{SL}}(2,\mathbb {C} ),n\geqslant 2,}$ ${\displaystyle \mathbf {P} _{n}^{2},}$

$${\displaystyle P{\begin{pmatrix}z_{1}\\z_{2}\\\end{pmatrix}}=c_{n}z_{1}^{n}+c_{n-1}z_{1}^{n-1}z_{2}+\cdots +c_{0}z_{2}^{n},\quad c_{0},c_{1},\ldots ,c_{n}\in \mathbb {Z} .}$$

La acción de está dada por ^{[80] [81]} ${\displaystyle {\text{SL}}(2,\mathbb {C} )}$

La -acción asociada es, utilizando (G6) y la definición anterior, para los elementos base de ^[82] ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )}$ ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} ),}$

Con una elección de base para , estas representaciones se convierten en álgebras de Lie matriciales. ${\displaystyle P\in \mathbf {P} _{n}^{2}}$

Representaciones lineales reales

Las representaciones ( μ , ν ) se realizan en un espacio de polinomios en homogéneos de grado μ en y homogéneos de grado ν en ^[79] Las representaciones están dadas por ^[83] ${\displaystyle \mathbf {P} _{\mu ,\nu }^{2}}$ ${\displaystyle z_{1},{\overline {z_{1}}},z_{2},{\overline {z_{2}}},}$ ${\displaystyle z_{1},z_{2}}$ ${\displaystyle {\overline {z_{1}}},{\overline {z_{2}}}.}$

Empleando (G6) nuevamente se encuentra que

En particular para los elementos básicos,

Propiedades de la (metro, norte) representaciones

Las representaciones ( m ,  n ) , definidas anteriormente a través de (A1) (como restricciones a la forma real ) de los productos tensoriales de las representaciones lineales complejas irreducibles π _{m = μ} y π _{n = ν} de son irreducibles, y son las únicas representaciones irreducibles. ^[61] ${\displaystyle {\mathfrak {sl}}(3,1)}$ ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} ),}$

• La irreductibilidad se deriva del truco unitario ^[84] y que una representación Π de SU(2) × SU(2) es irreducible si y sólo si Π = Π _μ ⊗ Π _ν , ^{[nb 23]} donde Π _μ , Π _ν son irreducibles representaciones de SU(2) .
• La unicidad se deduce de que los Π _m son las únicas representaciones irreducibles de SU(2) , lo que es una de las conclusiones del teorema de mayor peso. ^[85]

Dimensión

The (m, n) representations are (2m + 1)(2n + 1)-dimensional.^[86] This follows easiest from counting the dimensions in any concrete realization, such as the one given in representations of SL ( 2 , C ) {\displaystyle {\text{SL}}(2,\mathbb {C} )} and s l ( 2 , C ) {\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )} . For a Lie general algebra ${\displaystyle {\mathfrak {g}}}$ the Weyl dimension formula,^[87] $${\displaystyle \dim \pi _{\rho }={\frac {\Pi _{\alpha \in R^{+}}\langle \alpha ,\rho +\delta \rangle }{\Pi _{\alpha \in R^{+}}\langle \alpha ,\delta \rangle }},}$$applies, where R⁺ is the set of positive roots, ρ is the highest weight, and δ is half the sum of the positive roots. The inner product ${\displaystyle \langle \cdot ,\cdot \rangle }$ is that of the Lie algebra ${\displaystyle {\mathfrak {g}},}$ invariant under the action of the Weyl group on ${\displaystyle {\mathfrak {h}}\subset {\mathfrak {g}},}$ the Cartan subalgebra. The roots (really elements of ${\displaystyle {\mathfrak {h}}^{*}}$) are via this inner product identified with elements of ${\displaystyle {\mathfrak {h}}.}$ For ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} ),}$ the formula reduces to dim π_μ = 2μ + 1 = 2m + 1, where the present notation must be taken into account. The highest weight is 2μ.^[88] By taking tensor products, the result follows.

Faithfulness

If a representation Π of a Lie group G is not faithful, then N = ker Π is a nontrivial normal subgroup.^[89] There are three relevant cases.

1. N is non-discrete and abelian.
2. N is non-discrete and non-abelian.
3. N is discrete. In this case N ⊂ Z, where Z is the center of G.^{[nb 24]}

In the case of SO(3; 1)⁺, the first case is excluded since SO(3; 1)⁺ is semi-simple.^{[nb 25]} The second case (and the first case) is excluded because SO(3; 1)⁺ is simple.^{[nb 26]} For the third case, SO(3; 1)⁺ is isomorphic to the quotient ${\displaystyle {\text{SL}}(2,\mathbb {C} )/\{\pm I\}.}$ But ${\displaystyle \{\pm I\}}$ is the center of ${\displaystyle {\text{SL}}(2,\mathbb {C} ).}$ It follows that the center of SO(3; 1)⁺ is trivial, and this excludes the third case. The conclusion is that every representation Π : SO(3; 1)⁺ → GL(V) and every projective representation Π : SO(3; 1)⁺ → PGL(W) for V, W finite-dimensional vector spaces are faithful.

By using the fundamental Lie correspondence, the statements and the reasoning above translate directly to Lie algebras with (abelian) nontrivial non-discrete normal subgroups replaced by (one-dimensional) nontrivial ideals in the Lie algebra,^[90] and the center of SO(3; 1)⁺ replaced by the center of ${\displaystyle {\mathfrak {sl}}(3;1)^{+}}$The center of any semisimple Lie algebra is trivial^[91] and ${\displaystyle {\mathfrak {so}}(3;1)}$ is semi-simple and simple, and hence has no non-trivial ideals.

A related fact is that if the corresponding representation of ${\displaystyle {\text{SL}}(2,\mathbb {C} )}$ is faithful, then the representation is projective. Conversely, if the representation is non-projective, then the corresponding ${\displaystyle {\text{SL}}(2,\mathbb {C} )}$ representation is not faithful, but is 2:1.

Non-unitarity

The (m, n) Lie algebra representation is not Hermitian. Accordingly, the corresponding (projective) representation of the group is never unitary.^{[nb 27]} This is due to the non-compactness of the Lorentz group. In fact, a connected simple non-compact Lie group cannot have any nontrivial unitary finite-dimensional representations.^[33] There is a topological proof of this.^[92] Let u : G → GL(V), where V is finite-dimensional, be a continuous unitary representation of the non-compact connected simple Lie group G. Then u(G) ⊂ U(V) ⊂ GL(V) where U(V) is the compact subgroup of GL(V) consisting of unitary transformations of V. The kernel of u is a normal subgroup of G. Since G is simple, ker u is either all of G, in which case u is trivial, or ker u is trivial, in which case u is faithful. In the latter case u is a diffeomorphism onto its image,^[93] u(G) ≅ G and u(G) is a Lie group. This would mean that u(G) is an embedded non-compact Lie subgroup of the compact group U(V). This is impossible with the subspace topology on u(G) ⊂ U(V) since all embedded Lie subgroups of a Lie group are closed^[94] If u(G) were closed, it would be compact,^{[nb 28]} and then G would be compact,^{[nb 29]} contrary to assumption.^{[nb 30]}

In the case of the Lorentz group, this can also be seen directly from the definitions. The representations of A and B used in the construction are Hermitian. This means that J is Hermitian, but K is anti-Hermitian.^[95] The non-unitarity is not a problem in quantum field theory, since the objects of concern are not required to have a Lorentz-invariant positive definite norm.^[96]

Restriction to SO(3)

The (m, n) representation is, however, unitary when restricted to the rotation subgroup SO(3), but these representations are not irreducible as representations of SO(3). A Clebsch–Gordan decomposition can be applied showing that an (m, n) representation have SO(3)-invariant subspaces of highest weight (spin) m + n, m + n − 1, ..., | m − n|,^[97] where each possible highest weight (spin) occurs exactly once. A weight subspace of highest weight (spin) j is (2j + 1)-dimensional. So for example, the (⁠1/2⁠, ⁠1/2⁠) representation has spin 1 and spin 0 subspaces of dimension 3 and 1 respectively.

Since the angular momentum operator is given by J = A + B, the highest spin in quantum mechanics of the rotation sub-representation will be (m + n)ℏ and the "usual" rules of addition of angular momenta and the formalism of 3-j symbols, 6-j symbols, etc. applies.^[98]

Spinors

It is the SO(3)-invariant subspaces of the irreducible representations that determine whether a representation has spin. From the above paragraph, it is seen that the (m, n) representation has spin if m + n is half-integer. The simplest are (⁠1/2⁠, 0) and (0, ⁠1/2⁠), the Weyl-spinors of dimension 2. Then, for example, (0, ⁠3/2⁠) and (1, ⁠1/2⁠) are a spin representations of dimensions 2⋅⁠3/2⁠ + 1 = 4 and (2 + 1)(2⋅⁠1/2⁠ + 1) = 6 respectively. According to the above paragraph, there are subspaces with spin both ⁠3/2⁠ and ⁠1/2⁠ in the last two cases, so these representations cannot likely represent a single physical particle which must be well-behaved under SO(3). It cannot be ruled out in general, however, that representations with multiple SO(3) subrepresentations with different spin can represent physical particles with well-defined spin. It may be that there is a suitable relativistic wave equation that projects out unphysical components, leaving only a single spin.^[99]

Construction of pure spin ⁠n/2⁠ representations for any n (under SO(3)) from the irreducible representations involves taking tensor products of the Dirac-representation with a non-spin representation, extraction of a suitable subspace, and finally imposing differential constraints.^[100]

Dual representations

The root system A₁ × A₁ of ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )\oplus {\mathfrak {sl}}(2,\mathbb {C} ).}$

The following theorems are applied to examine whether the dual representation of an irreducible representation is isomorphic to the original representation:

1. The set of weights of the dual representation of an irreducible representation of a semisimple Lie algebra is, including multiplicities, the negative of the set of weights for the original representation.^[101]
2. Two irreducible representations are isomorphic if and only if they have the same highest weight.^{[nb 31]}
3. For each semisimple Lie algebra there exists a unique element w₀ of the Weyl group such that if μ is a dominant integral weight, then w₀ ⋅ (−μ) is again a dominant integral weight.^[102]
4. If ${\displaystyle \pi _{\mu _{0}}}$ is an irreducible representation with highest weight μ₀, then ${\displaystyle \pi _{\mu _{0}}^{*}}$ has highest weight w₀ ⋅ (−μ).^[102]

Here, the elements of the Weyl group are considered as orthogonal transformations, acting by matrix multiplication, on the real vector space of roots. If −I is an element of the Weyl group of a semisimple Lie algebra, then w₀ = −I. In the case of ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} ),}$ the Weyl group is W = {I, −I}.^[103] It follows that each π_μ, μ = 0, 1, ... is isomorphic to its dual ${\displaystyle \pi _{\mu }^{*}.}$ The root system of ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )\oplus {\mathfrak {sl}}(2,\mathbb {C} )}$ is shown in the figure to the right.^{[nb 32]} The Weyl group is generated by ${\displaystyle \{w_{\gamma }\}}$ where ${\displaystyle w_{\gamma }}$ is reflection in the plane orthogonal to γ as γ ranges over all roots.^{[nb 33]} Inspection shows that w_α ⋅ w_β = −I so −I ∈ W. Using the fact that if π, σ are Lie algebra representations and π ≅ σ, then Π ≅ Σ,^[104] the conclusion for SO(3; 1)⁺ is $${\displaystyle \pi _{m,n}^{*}\cong \pi _{m,n},\quad \Pi _{m,n}^{*}\cong \Pi _{m,n},\quad 2m,2n\in \mathbf {N} .}$$

Complex conjugate representations

If π is a representation of a Lie algebra, then ${\displaystyle {\overline {\pi }}}$ is a representation, where the bar denotes entry-wise complex conjugation in the representative matrices. This follows from that complex conjugation commutes with addition and multiplication.^[105] In general, every irreducible representation π of ${\displaystyle {\mathfrak {sl}}(n,\mathbb {C} )}$ can be written uniquely as π = π⁺ + π⁻, where^[106] $${\displaystyle \pi ^{\pm }(X)={\frac {1}{2}}\left(\pi (X)\pm i\pi \left(i^{-1}X\right)\right),}$$with ${\displaystyle \pi ^{+}}$ holomorphic (complex linear) and ${\displaystyle \pi ^{-}}$ anti-holomorphic (conjugate linear). For ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} ),}$ since ${\displaystyle \pi _{\mu }}$ is holomorphic, ${\displaystyle {\overline {\pi _{\mu }}}}$ is anti-holomorphic. Direct examination of the explicit expressions for ${\displaystyle \pi _{\mu ,0}}$ and ${\displaystyle \pi _{0,\nu }}$ in equation (S8) below shows that they are holomorphic and anti-holomorphic respectively. Closer examination of the expression (S8) also allows for identification of ${\displaystyle \pi ^{+}}$ and ${\displaystyle \pi ^{-}}$ for ${\displaystyle \pi _{\mu ,\nu }}$ as $${\displaystyle \pi _{\mu ,\nu }^{+}=\pi _{\mu }^{\oplus _{\nu +1}},\qquad \pi _{\mu ,\nu }^{-}={\overline {\pi _{\nu }^{\oplus _{\mu +1}}}}.}$$

Using the above identities (interpreted as pointwise addition of functions), for SO(3; 1)⁺ yields $${\displaystyle {\begin{aligned}{\overline {\pi _{m,n}}}&={\overline {\pi _{m,n}^{+}+\pi _{m,n}^{-}}}={\overline {\pi _{m}^{\oplus _{2n+1}}}}+{\overline {{\overline {\pi _{n}}}^{\oplus _{2m+1}}}}\\&=\pi _{n}^{\oplus _{2m+1}}+{\overline {\pi _{m}}}^{\oplus _{2n+1}}=\pi _{n,m}^{+}+\pi _{n,m}^{-}=\pi _{n,m}\\&&&2m,2n\in \mathbb {N} \\{\overline {\Pi _{m,n}}}&=\Pi _{n,m}\end{aligned}}}$$where the statement for the group representations follow from exp(X) = exp(X). It follows that the irreducible representations (m, n) have real matrix representatives if and only if m = n. Reducible representations on the form (m, n) ⊕ (n, m) have real matrices too.

The adjoint representation, the Clifford algebra, and the Dirac spinor representation

Richard Brauer and wife Ilse 1970. Brauer generalized the spin representations of Lie algebras sitting inside Clifford algebras to spin higher than ⁠1/2⁠.

Photo courtesy of MFO.

In general representation theory, if (π, V) is a representation of a Lie algebra ${\displaystyle {\mathfrak {g}},}$ then there is an associated representation of ${\displaystyle {\mathfrak {g}},}$ on End(V), also denoted π, given by

Likewise, a representation (Π, V) of a group G yields a representation Π on End(V) of G, still denoted Π, given by^[107]

If π and Π are the standard representations on ${\displaystyle \mathbb {R} ^{4}}$ and if the action is restricted to ${\displaystyle {\mathfrak {so}}(3,1)\subset {\text{End}}(\mathbb {R} ^{4}),}$ then the two above representations are the adjoint representation of the Lie algebra and the adjoint representation of the group respectively. The corresponding representations (some ${\displaystyle \mathbb {R} ^{n}}$ or ${\displaystyle \mathbb {C} ^{n}}$) always exist for any matrix Lie group, and are paramount for investigation of the representation theory in general, and for any given Lie group in particular.

Applying this to the Lorentz group, if (Π, V) is a projective representation, then direct calculation using (G5) shows that the induced representation on End(V) is a proper representation, i.e. a representation without phase factors.

In quantum mechanics this means that if (π, H) or (Π, H) is a representation acting on some Hilbert space H, then the corresponding induced representation acts on the set of linear operators on H. As an example, the induced representation of the projective spin (⁠1/2⁠, 0) ⊕ (0, ⁠1/2⁠) representation on End(H) is the non-projective 4-vector (⁠1/2⁠, ⁠1/2⁠) representation.^[108]

For simplicity, consider only the "discrete part" of End(H), that is, given a basis for H, the set of constant matrices of various dimension, including possibly infinite dimensions. The induced 4-vector representation of above on this simplified End(H) has an invariant 4-dimensional subspace that is spanned by the four gamma matrices.^[109] (The metric convention is different in the linked article.) In a corresponding way, the complete Clifford algebra of spacetime, ${\displaystyle {\mathcal {Cl}}_{3,1}(\mathbb {R} ),}$ whose complexification is ${\displaystyle {\text{M}}(4,\mathbb {C} ),}$ generated by the gamma matrices decomposes as a direct sum of representation spaces of a scalar irreducible representation (irrep), the (0, 0), a pseudoscalar irrep, also the (0, 0), but with parity inversion eigenvalue −1, see the next section below, the already mentioned vector irrep, (⁠1/2⁠, ⁠1/2⁠), a pseudovector irrep, (⁠1/2⁠, ⁠1/2⁠) with parity inversion eigenvalue +1 (not −1), and a tensor irrep, (1, 0) ⊕ (0, 1).^[110] The dimensions add up to 1 + 1 + 4 + 4 + 6 = 16. In other words,

where, as is customary, a representation is confused with its representation space.

The (⁠1/2⁠, 0) ⊕ (0, ⁠1/2⁠) spin representation

The six-dimensional representation space of the tensor (1, 0) ⊕ (0, 1)-representation inside ${\displaystyle {\mathcal {Cl}}_{3,1}(\mathbb {R} )}$ has two roles. The^[111]

where ${\displaystyle \gamma ^{0},\ldots ,\gamma ^{3}\in {\mathcal {Cl}}_{3,1}(\mathbb {R} )}$ are the gamma matrices, the sigmas, only 6 of which are non-zero due to antisymmetry of the bracket, span the tensor representation space. Moreover, they have the commutation relations of the Lorentz Lie algebra,^[112]

and hence constitute a representation (in addition to spanning a representation space) sitting inside ${\displaystyle {\mathcal {Cl}}_{3,1}(\mathbb {R} ),}$ the (⁠1/2⁠, 0) ⊕ (0, ⁠1/2⁠) spin representation. For details, see bispinor and Dirac algebra.

The conclusion is that every element of the complexified ${\displaystyle {\mathcal {Cl}}_{3,1}(\mathbb {R} )}$ in End(H) (i.e. every complex 4×4 matrix) has well defined Lorentz transformation properties. In addition, it has a spin-representation of the Lorentz Lie algebra, which upon exponentiation becomes a spin representation of the group, acting on ${\displaystyle \mathbb {C} ^{4},}$ making it a space of bispinors.

Reducible representations

There is a multitude of other representations that can be deduced from the irreducible ones, such as those obtained by taking direct sums, tensor products, and quotients of the irreducible representations. Other methods of obtaining representations include the restriction of a representation of a larger group containing the Lorentz group, e.g. ${\displaystyle {\text{GL}}(n,\mathbb {R} )}$ and the Poincaré group. These representations are in general not irreducible.

The Lorentz group and its Lie algebra have the complete reducibility property. This means that every representation reduces to a direct sum of irreducible representations. The reducible representations will therefore not be discussed.

Space inversion and time reversal

The (possibly projective) (m, n) representation is irreducible as a representation SO(3; 1)⁺, the identity component of the Lorentz group, in physics terminology the proper orthochronous Lorentz group. If m = n it can be extended to a representation of all of O(3; 1), the full Lorentz group, including space parity inversion and time reversal. The representations (m, n) ⊕ (n, m) can be extended likewise.^[113]

Space parity inversion

For space parity inversion, the adjoint action Ad_P of P ∈ SO(3; 1) on ${\displaystyle {\mathfrak {so}}(3;1)}$ is considered, where P is the standard representative of space parity inversion, P = diag(1, −1, −1, −1), given by

It is these properties of K and J under P that motivate the terms vector for K and pseudovector or axial vector for J. In a similar way, if π is any representation of ${\displaystyle {\mathfrak {so}}(3;1)}$ and Π is its associated group representation, then Π(SO(3; 1)⁺) acts on the representation of π by the adjoint action, π(X) ↦ Π(g) π(X) Π(g)⁻¹ for ${\displaystyle X\in {\mathfrak {so}}(3;1),}$ g ∈ SO(3; 1)⁺. If P is to be included in Π, then consistency with (F1) requires that

holds, where A and B are defined as in the first section. This can hold only if A_i and B_i have the same dimensions, i.e. only if m = n. When m ≠ n then (m, n) ⊕ (n, m) can be extended to an irreducible representation of SO(3; 1)⁺, the orthochronous Lorentz group. The parity reversal representative Π(P) does not come automatically with the general construction of the (m, n) representations. It must be specified separately. The matrix β = i γ⁰ (or a multiple of modulus −1 times it) may be used in the (⁠1/2⁠, 0) ⊕ (0, ⁠1/2⁠)^[114] representation.

If parity is included with a minus sign (the 1×1 matrix [−1]) in the (0,0) representation, it is called a pseudoscalar representation.

Time reversal

Time reversal T = diag(−1, 1, 1, 1), acts similarly on ${\displaystyle {\mathfrak {so}}(3;1)}$ by^[115]

By explicitly including a representative for T, as well as one for P, a representation of the full Lorentz group O(3; 1) is obtained. A subtle problem appears however in application to physics, in particular quantum mechanics. When considering the full Poincaré group, four more generators, the P^μ, in addition to the Jⁱ and Kⁱ generate the group. These are interpreted as generators of translations. The time-component P⁰ is the Hamiltonian H. The operator T satisfies the relation^[116]

in analogy to the relations above with ${\displaystyle {\mathfrak {so}}(3;1)}$ replaced by the full Poincaré algebra. By just cancelling the i's, the result THT⁻¹ = −H would imply that for every state Ψ with positive energy E in a Hilbert space of quantum states with time-reversal invariance, there would be a state Π(T⁻¹)Ψ with negative energy −E. Such states do not exist. The operator Π(T) is therefore chosen antilinear and antiunitary, so that it anticommutes with i, resulting in THT⁻¹ = H, and its action on Hilbert space likewise becomes antilinear and antiunitary.^[117] It may be expressed as the composition of complex conjugation with multiplication by a unitary matrix.^[118] This is mathematically sound, see Wigner's theorem, but with very strict requirements on terminology, Π is not a representation.

When constructing theories such as QED which is invariant under space parity and time reversal, Dirac spinors may be used, while theories that do not, such as the electroweak force, must be formulated in terms of Weyl spinors. The Dirac representation, (⁠1/2⁠, 0) ⊕ (0, ⁠1/2⁠), is usually taken to include both space parity and time inversions. Without space parity inversion, it is not an irreducible representation.

The third discrete symmetry entering in the CPT theorem along with P and T, charge conjugation symmetry C, has nothing directly to do with Lorentz invariance.^[119]

Action on function spaces

If V is a vector space of functions of a finite number of variables n, then the action on a scalar function ${\displaystyle f\in V}$ given by

produces another function Πf ∈ V. Here Π_x is an n-dimensional representation, and Π is a possibly infinite-dimensional representation. A special case of this construction is when V is a space of functions defined on the a linear group G itself, viewed as a n-dimensional manifold embedded in ${\displaystyle \mathbb {R} ^{m^{2}}}$ (with m the dimension of the matrices).^[120] This is the setting in which the Peter–Weyl theorem and the Borel–Weil theorem are formulated. The former demonstrates the existence of a Fourier decomposition of functions on a compact group into characters of finite-dimensional representations.^[61] The latter theorem, providing more explicit representations, makes use of the unitarian trick to yield representations of complex non-compact groups, e.g. ${\displaystyle {\text{SL}}(2,\mathbb {C} ).}$

The following exemplifies action of the Lorentz group and the rotation subgroup on some function spaces.

Euclidean rotations

The subgroup SO(3) of three-dimensional Euclidean rotations has an infinite-dimensional representation on the Hilbert space $${\displaystyle L^{2}\left(\mathbb {S} ^{2}\right)=\operatorname {span} \left\{Y_{m}^{l},l\in \mathbb {N} ^{+},-l\leqslant m\leqslant l\right\},}$$

where ${\displaystyle Y_{m}^{l}}$ are the spherical harmonics. An arbitrary square integrable function f on the unit sphere can be expressed as^[121]

where the f_lm are generalized Fourier coefficients.

The Lorentz group action restricts to that of SO(3) and is expressed as

where the D^l are obtained from the representatives of odd dimension of the generators of rotation.

The Möbius group

The identity component of the Lorentz group is isomorphic to the Möbius group M. This group can be thought of as conformal mappings of either the complex plane or, via stereographic projection, the Riemann sphere. In this way, the Lorentz group itself can be thought of as acting conformally on the complex plane or on the Riemann sphere.

In the plane, a Möbius transformation characterized by the complex numbers a, b, c, d acts on the plane according to^[122]

and can be represented by complex matrices

since multiplication by a nonzero complex scalar does not change f. These are elements of ${\displaystyle {\text{SL}}(2,\mathbb {C} )}$ and are unique up to a sign (since ±Π_f give the same f), hence ${\displaystyle {\text{SL}}(2,\mathbb {C} )/\{\pm I\}\cong {\text{SO}}(3;1)^{+}.}$

The Riemann P-functions

The Riemann P-functions, solutions of Riemann's differential equation, are an example of a set of functions that transform among themselves under the action of the Lorentz group. The Riemann P-functions are expressed as^[123]

where the a,  b,  c,  α,  β,  γ,  α′,  β′,  γ′ are complex constants. The P-function on the right hand side can be expressed using standard hypergeometric functions. The connection is^[124]

The set of constants 0, ∞, 1 in the upper row on the left hand side are the regular singular points of the Gauss' hypergeometric equation.^[125] Its exponents, i. e. solutions of the indicial equation, for expansion around the singular point 0 are 0 and 1 − c ,corresponding to the two linearly independent solutions,^{[nb 34]} and for expansion around the singular point 1 they are 0 and c − a − b.^[126] Similarly, the exponents for ∞ are a and b for the two solutions.^[127]

One has thus

where the condition (sometimes called Riemann's identity)^[128] $${\displaystyle \alpha +\alpha '+\beta +\beta '+\gamma +\gamma '=1}$$on the exponents of the solutions of Riemann's differential equation has been used to define γ′.

The first set of constants on the left hand side in (T1), a, b, c denotes the regular singular points of Riemann's differential equation. The second set, α, β, γ, are the corresponding exponents at a, b, c for one of the two linearly independent solutions, and, accordingly, α′, β′, γ′ are exponents at a, b, c for the second solution.

Define an action of the Lorentz group on the set of all Riemann P-functions by first setting

where A,  B,  C,  D are the entries in

for Λ = p(λ) ∈ SO(3; 1)⁺ a Lorentz transformation.

Define

where P is a Riemann P-function. The resulting function is again a Riemann P-function. The effect of the Möbius transformation of the argument is that of shifting the poles to new locations, hence changing the critical points, but there is no change in the exponents of the differential equation the new function satisfies. The new function is expressed as

where

Infinite-dimensional unitary representations

History

The Lorentz group SO(3; 1)⁺ and its double cover ${\displaystyle {\text{SL}}(2,\mathbb {C} )}$ also have infinite dimensional unitary representations, studied independently by Bargmann (1947), Gelfand & Naimark (1947) and Harish-Chandra (1947) at the instigation of Paul Dirac.^[129][130] This trail of development begun with Dirac (1936) where he devised matrices U and B necessary for description of higher spin (compare Dirac matrices), elaborated upon by Fierz (1939), see also Fierz & Pauli (1939), and proposed precursors of the Bargmann-Wigner equations.^[131] In Dirac (1945) he proposed a concrete infinite-dimensional representation space whose elements were called expansors as a generalization of tensors.^{[nb 35]} These ideas were incorporated by Harish–Chandra and expanded with expinors as an infinite-dimensional generalization of spinors in his 1947 paper.

The Plancherel formula for these groups was first obtained by Gelfand and Naimark through involved calculations. The treatment was subsequently considerably simplified by Harish-Chandra (1951) and Gelfand & Graev (1953), based on an analogue for ${\displaystyle {\text{SL}}(2,\mathbb {C} )}$ of the integration formula of Hermann Weyl for compact Lie groups.^[132] Elementary accounts of this approach can be found in Rühl (1970) and Knapp (2001).

The theory of spherical functions for the Lorentz group, required for harmonic analysis on the hyperboloid model of 3-dimensional hyperbolic space sitting in Minkowski space is considerably easier than the general theory. It only involves representations from the spherical principal series and can be treated directly, because in radial coordinates the Laplacian on the hyperboloid is equivalent to the Laplacian on ${\displaystyle \mathbb {R} .}$ This theory is discussed in Takahashi (1963), Helgason (1968), Helgason (2000) and the posthumous text of Jorgenson & Lang (2008).

Principal series for SL(2, C)

The principal series, or unitary principal series, are the unitary representations induced from the one-dimensional representations of the lower triangular subgroup B of ${\displaystyle G={\text{SL}}(2,\mathbb {C} ).}$ Since the one-dimensional representations of B correspond to the representations of the diagonal matrices, with non-zero complex entries z and z⁻¹, they thus have the form $${\displaystyle \chi _{\nu ,k}{\begin{pmatrix}z&0\\c&z^{-1}\end{pmatrix}}=r^{i\nu }e^{ik\theta },}$$for k an integer, ν real and with z = re^iθ. The representations are irreducible; the only repetitions, i.e. isomorphisms of representations, occur when k is replaced by −k. By definition the representations are realized on L² sections of line bundles on ${\displaystyle G/B=\mathbb {S} ^{2},}$ which is isomorphic to the Riemann sphere. When k = 0, these representations constitute the so-called spherical principal series.

The restriction of a principal series to the maximal compact subgroup K = SU(2) of G can also be realized as an induced representation of K using the identification G/B = K/T, where T = B ∩ K is the maximal torus in K consisting of diagonal matrices with | z | = 1. It is the representation induced from the 1-dimensional representation z^kT, and is independent of ν. By Frobenius reciprocity, on K they decompose as a direct sum of the irreducible representations of K with dimensions |k| + 2m + 1 with m a non-negative integer.

Using the identification between the Riemann sphere minus a point and ${\displaystyle \mathbb {C} ,}$ the principal series can be defined directly on ${\displaystyle L^{2}(\mathbb {C} )}$ by the formula^[133] $${\displaystyle \pi _{\nu ,k}{\begin{pmatrix}a&b\\c&d\end{pmatrix}}^{-1}f(z)=|cz+d|^{-2-i\nu }\left({cz+d \over |cz+d|}\right)^{-k}f\left({az+b \over cz+d}\right).}$$

Irreducibility can be checked in a variety of ways:

• The representation is already irreducible on B. This can be seen directly, but is also a special case of general results on irreducibility of induced representations due to François Bruhat and George Mackey, relying on the Bruhat decomposition G = B ∪ BsB where s is the Weyl group element^[134] $${\displaystyle {\begin{pmatrix}0&-1\\1&0\end{pmatrix}}}$$.
• The action of the Lie algebra ${\displaystyle {\mathfrak {g}}}$ of G can be computed on the algebraic direct sum of the irreducible subspaces of K can be computed explicitly and the it can be verified directly that the lowest-dimensional subspace generates this direct sum as a ${\displaystyle {\mathfrak {g}}}$-module.^[8][135]

Complementary series for SL(2, C)

The for 0 < t < 2, the complementary series is defined on ${\displaystyle L^{2}(\mathbb {C} )}$ for the inner product^[136] $${\displaystyle (f,g)_{t}=\iint {\frac {f(z){\overline {g(w)}}}{|z-w|^{2-t}}}\,dz\,dw,}$$with the action given by^[137][138] $${\displaystyle \pi _{t}{\begin{pmatrix}a&b\\c&d\end{pmatrix}}^{-1}f(z)=|cz+d|^{-2-t}f\left({az+b \over cz+d}\right).}$$

The representations in the complementary series are irreducible and pairwise non-isomorphic. As a representation of K, each is isomorphic to the Hilbert space direct sum of all the odd dimensional irreducible representations of K = SU(2). Irreducibility can be proved by analyzing the action of ${\displaystyle {\mathfrak {g}}}$ on the algebraic sum of these subspaces^[8][135] or directly without using the Lie algebra.^[139][140]

Plancherel theorem for SL(2, C)

The only irreducible unitary representations of ${\displaystyle {\text{SL}}(2,\mathbb {C} )}$ are the principal series, the complementary series and the trivial representation. Since −I acts as (−1)^k on the principal series and trivially on the remainder, these will give all the irreducible unitary representations of the Lorentz group, provided k is taken to be even.

To decompose the left regular representation of G on ${\displaystyle L^{2}(G)}$ only the principal series are required. This immediately yields the decomposition on the subrepresentations ${\displaystyle L^{2}(G/\{\pm I\}),}$ the left regular representation of the Lorentz group, and ${\displaystyle L^{2}(G/K),}$ the regular representation on 3-dimensional hyperbolic space. (The former only involves principal series representations with k even and the latter only those with k = 0.)

The left and right regular representation λ and ρ are defined on ${\displaystyle L^{2}(G)}$ by $${\displaystyle {\begin{aligned}(\lambda (g)f)(x)&=f\left(g^{-1}x\right)\\(\rho (g)f)(x)&=f(xg)\end{aligned}}}$$

Now if f is an element of C_c(G), the operator ${\displaystyle \pi _{\nu ,k}(f)}$ defined by $${\displaystyle \pi _{\nu ,k}(f)=\int _{G}f(g)\pi (g)\,dg}$$is Hilbert–Schmidt. Define a Hilbert space H by $${\displaystyle H=\bigoplus _{k\geqslant 0}{\text{HS}}\left(L^{2}(\mathbb {C} )\right)\otimes L^{2}\left(\mathbb {R} ,c_{k}{\sqrt {\nu ^{2}+k^{2}}}d\nu \right),}$$where $${\displaystyle c_{k}={\begin{cases}{\frac {1}{4\pi ^{3/2}}}&k=0\\{\frac {1}{(2\pi )^{3/2}}}&k\neq 0\end{cases}}}$$and ${\displaystyle {\text{HS}}\left(L^{2}(\mathbb {C} )\right)}$ denotes the Hilbert space of Hilbert–Schmidt operators on ${\displaystyle L^{2}(\mathbb {C} ).}$^{[nb 36]} Then the map U defined on C_c(G) by $${\displaystyle U(f)(\nu ,k)=\pi _{\nu ,k}(f)}$$extends to a unitary of ${\displaystyle L^{2}(G)}$ onto H.

The map U satisfies the intertwining property $${\displaystyle U(\lambda (x)\rho (y)f)(\nu ,k)=\pi _{\nu ,k}(x)^{-1}\pi _{\nu ,k}(f)\pi _{\nu ,k}(y).}$$

If f₁, f₂ are in C_c(G) then by unitarity $${\displaystyle (f_{1},f_{2})=\sum _{k\geqslant 0}c_{k}^{2}\int _{-\infty }^{\infty }\operatorname {Tr} \left(\pi _{\nu ,k}(f_{1})\pi _{\nu ,k}(f_{2})^{*}\right)\left(\nu ^{2}+k^{2}\right)\,d\nu .}$$

Thus if ${\displaystyle f=f_{1}*f_{2}^{*}}$ denotes the convolution of ${\displaystyle f_{1}}$ and ${\displaystyle f_{2}^{*},}$ and ${\displaystyle f_{2}^{*}(g)={\overline {f_{2}(g^{-1})}},}$ then^[141] $${\displaystyle f(1)=\sum _{k\geqslant 0}c_{k}^{2}\int _{-\infty }^{\infty }\operatorname {Tr} \left(\pi _{\nu ,k}(f)\right)\left(\nu ^{2}+k^{2}\right)\,d\nu .}$$

The last two displayed formulas are usually referred to as the Plancherel formula and the Fourier inversion formula respectively.

The Plancherel formula extends to all ${\displaystyle f_{i}\in L^{2}(G).}$ By a theorem of Jacques Dixmier and Paul Malliavin, every smooth compactly supported function on ${\displaystyle G}$ is a finite sum of convolutions of similar functions, the inversion formula holds for such f. It can be extended to much wider classes of functions satisfying mild differentiability conditions.^[61]

Classification of representations of SO(3, 1)

The strategy followed in the classification of the irreducible infinite-dimensional representations is, in analogy to the finite-dimensional case, to assume they exist, and to investigate their properties. Thus first assume that an irreducible strongly continuous infinite-dimensional representation Π_H on a Hilbert space H of SO(3; 1)⁺ is at hand.^[142] Since SO(3) is a subgroup, Π_H is a representation of it as well. Each irreducible subrepresentation of SO(3) is finite-dimensional, and the SO(3) representation is reducible into a direct sum of irreducible finite-dimensional unitary representations of SO(3) if Π_H is unitary.^[143]

The steps are the following:^[144]

1. Choose a suitable basis of common eigenvectors of J² and J₃.
2. Compute matrix elements of J₁, J₂, J₃ and K₁, K₂, K₃.
3. Enforce Lie algebra commutation relations.
4. Require unitarity together with orthonormality of the basis.^{[nb 37]}

Step 1

One suitable choice of basis and labeling is given by $${\displaystyle \left|j_{0}\,j_{1};j\,m\right\rangle .}$$

If this were a finite-dimensional representation, then j₀ would correspond the lowest occurring eigenvalue j(j + 1) of J² in the representation, equal to |m − n|, and j₁ would correspond to the highest occurring eigenvalue, equal to m + n. In the infinite-dimensional case, j₀ ≥ 0 retains this meaning, but j₁ does not.^[66] For simplicity, it is assumed that a given j occurs at most once in a given representation (this is the case for finite-dimensional representations), and it can be shown^[145] that the assumption is possible to avoid (with a slightly more complicated calculation) with the same results.

Step 2

The next step is to compute the matrix elements of the operators J₁, J₂, J₃ and K₁, K₂, K₃ forming the basis of the Lie algebra of ${\displaystyle {\mathfrak {so}}(3;1).}$ The matrix elements of ${\displaystyle J_{\pm }=J_{1}\pm iJ_{2}}$ and ${\displaystyle J_{3}}$ (the complexified Lie algebra is understood) are known from the representation theory of the rotation group, and are given by^[146][147] $${\displaystyle {\begin{aligned}\left\langle j\,m\right|J_{+}\left|j\,m-1\right\rangle =\left\langle j\,m-1\right|J_{-}\left|j\,m\right\rangle &={\sqrt {(j+m)(j-m+1)}},\\\left\langle j\,m\right|J_{3}\left|j\,m\right\rangle &=m,\end{aligned}}}$$where the labels j₀ and j₁ have been dropped since they are the same for all basis vectors in the representation.

Due to the commutation relations $${\displaystyle [J_{i},K_{j}]=i\epsilon _{ijk}K_{k},}$$ the triple (K₁, K₂, K₃) ≡ K is a vector operator^[148] and the Wigner–Eckart theorem^[149] applies for computation of matrix elements between the states represented by the chosen basis.^[150] The matrix elements of $${\displaystyle {\begin{aligned}K_{0}^{(1)}&=K_{3},\\K_{\pm 1}^{(1)}&=\mp {\frac {1}{\sqrt {2}}}(K_{1}\pm iK_{2}),\end{aligned}}}$$

where the superscript (1) signifies that the defined quantities are the components of a spherical tensor operator of rank k = 1 (which explains the factor √2 as well) and the subscripts 0, ±1 are referred to as q in formulas below, are given by^[151] $${\displaystyle {\begin{aligned}\left\langle j'm'\left|K_{0}^{(1)}\right|j\,m\right\rangle &=\left\langle j'\,m'\,k=1\,q=0|j\,m\right\rangle \left\langle j\left\|K^{(1)}\right\|j'\right\rangle ,\\\left\langle j'm'\left|K_{\pm 1}^{(1)}\right|j\,m\right\rangle &=\left\langle j'\,m'\,k=1\,q=\pm 1|j\,m\right\rangle \left\langle j\left\|K^{(1)}\right\|j'\right\rangle .\end{aligned}}}$$

Here the first factors on the right hand sides are Clebsch–Gordan coefficients for coupling j′ with k to get j. The second factors are the reduced matrix elements. They do not depend on m, m′ or q, but depend on j, j′ and, of course, K. For a complete list of non-vanishing equations, see Harish-Chandra (1947, p. 375).

Step 3

The next step is to demand that the Lie algebra relations hold, i.e. that $${\displaystyle [K_{\pm },K_{3}]=\pm J_{\pm },\quad [K_{+},K_{-}]=-2J_{3}.}$$

This results in a set of equations^[152] for which the solutions are^[153] $${\displaystyle {\begin{aligned}\left\langle j\left\|K^{(1)}\right\|j\right\rangle &=i{\frac {j_{1}j_{0}}{\sqrt {j(j+1)}}},\\\left\langle j\left\|K^{(1)}\right\|j-1\right\rangle &=-B_{j}\xi _{j}{\sqrt {j(2j-1)}},\\\left\langle j-1\left\|K^{(1)}\right\|j\right\rangle &=B_{j}\xi _{j}^{-1}{\sqrt {j(2j+1)}},\end{aligned}}}$$where $${\displaystyle B_{j}={\sqrt {\frac {(j^{2}-j_{0}^{2})(j^{2}-j_{1}^{2})}{j^{2}(4j^{2}-1)}}},\quad j_{0}=0,{\tfrac {1}{2}},1,\ldots \quad {\text{and}}\quad j_{1},\xi _{j}\in \mathbb {C} .}$$

Step 4

The imposition of the requirement of unitarity of the corresponding representation of the group restricts the possible values for the arbitrary complex numbers j₀ and ξ_j. Unitarity of the group representation translates to the requirement of the Lie algebra representatives being Hermitian, meaning $${\displaystyle K_{\pm }^{\dagger }=K_{\mp },\quad K_{3}^{\dagger }=K_{3}.}$$

This translates to^[154] $${\displaystyle {\begin{aligned}\left\langle j\left\|K^{(1)}\right\|j\right\rangle &={\overline {\left\langle j\left\|K^{(1)}\right\|j\right\rangle }},\\\left\langle j\left\|K^{(1)}\right\|j-1\right\rangle &=-{\overline {\left\langle j-1\left\|K^{(1)}\right\|j\right\rangle }},\end{aligned}}}$$leading to^[155] $${\displaystyle {\begin{aligned}j_{0}\left(j_{1}+{\overline {j_{1}}}\right)&=0,\\\left|B_{j}\right|\left(\left|\xi _{j}\right|^{2}-e^{-2i\beta _{j}}\right)&=0,\end{aligned}}}$$where β_j is the angle of B_j on polar form. For |B_j| ≠ 0 follows ${\displaystyle \left|\xi _{j}\right|^{2}=1}$ and ${\displaystyle \xi _{j}=1}$ is chosen by convention. There are two possible cases:

• ${\displaystyle {\underline {j_{1}+{\overline {j_{1}}}=0.}}}$ In this case j₁ = − iν, ν real,^[156] $${\displaystyle \left\langle j\left\|K^{(1)}\right\|j\right\rangle ={\frac {\nu j_{0}}{j(j+1)}}\quad {\text{and}}\quad B_{j}={\sqrt {\frac {(j^{2}-j_{0}^{2})(j^{2}+\nu ^{2})}{4j^{2}-1}}}}$$ This is the principal series. Its elements are denoted ${\displaystyle (j_{0},\nu ),2j_{0}\in \mathbb {N} ,\nu \in \mathbb {R} .}$
• ${\displaystyle {\underline {j_{0}=0.}}}$ It follows:^[157] $${\displaystyle \left\langle j\left\|K^{(1)}\right\|j\right\rangle =0\quad {\text{and}}\quad B_{j}={\sqrt {\frac {j^{2}-\nu ^{2}}{4j^{2}-1}}}}$$ Since B₀ = B_j0, B²
  _j is real and positive for j = 1, 2, ..., leading to −1 ≤ ν ≤ 1. This is complementary series. Its elements are denoted (0, ν), −1 ≤ ν ≤ 1

This shows that the representations of above are all infinite-dimensional irreducible unitary representations.

Explicit formulas

Conventions and Lie algebra bases

The metric of choice is given by η = diag(−1, 1, 1, 1), and the physics convention for Lie algebras and the exponential mapping is used. These choices are arbitrary, but once they are made, fixed. One possible choice of basis for the Lie algebra is, in the 4-vector representation, given by: $${\displaystyle {\begin{aligned}J_{1}=J^{23}=-J^{32}&=i{\begin{pmatrix}0&0&0&0\\0&0&0&0\\0&0&0&-1\\0&0&1&0\end{pmatrix}},&K_{1}=J^{01}=-J^{10}&=i{\begin{pmatrix}0&1&0&0\\1&0&0&0\\0&0&0&0\\0&0&0&0\end{pmatrix}},\\[8pt]J_{2}=J^{31}=-J^{13}&=i{\begin{pmatrix}0&0&0&0\\0&0&0&1\\0&0&0&0\\0&-1&0&0\end{pmatrix}},&K_{2}=J^{02}=-J^{20}&=i{\begin{pmatrix}0&0&1&0\\0&0&0&0\\1&0&0&0\\0&0&0&0\end{pmatrix}},\\[8pt]J_{3}=J^{12}=-J^{21}&=i{\begin{pmatrix}0&0&0&0\\0&0&-1&0\\0&1&0&0\\0&0&0&0\end{pmatrix}},&K_{3}=J^{03}=-J^{30}&=i{\begin{pmatrix}0&0&0&1\\0&0&0&0\\0&0&0&0\\1&0&0&0\end{pmatrix}}.\\[8pt]\end{aligned}}}$$

The commutation relations of the Lie algebra ${\displaystyle {\mathfrak {so}}(3;1)}$ are:^[158] $${\displaystyle \left[J^{\mu \nu },J^{\rho \sigma }\right]=i\left(\eta ^{\sigma \mu }J^{\rho \nu }+\eta ^{\nu \sigma }J^{\mu \rho }-\eta ^{\rho \mu }J^{\sigma \nu }-\eta ^{\nu \rho }J^{\mu \sigma }\right).}$$

In three-dimensional notation, these are^[159] $${\displaystyle \left[J_{i},J_{j}\right]=i\epsilon _{ijk}J_{k},\quad \left[J_{i},K_{j}\right]=i\epsilon _{ijk}K_{k},\quad \left[K_{i},K_{j}\right]=-i\epsilon _{ijk}J_{k}.}$$

The choice of basis above satisfies the relations, but other choices are possible. The multiple use of the symbol J above and in the sequel should be observed.

For example, a typical boost and a typical rotation exponentiate as, $${\displaystyle \exp(-i\xi K_{1})={\begin{pmatrix}\cosh \xi &\sinh \xi &0&0\\\sinh \xi &\cosh \xi &0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}},\qquad \exp(-i\theta J_{1})={\begin{pmatrix}1&0&0&0\\0&1&0&0\\0&0&\cos \theta &-\sin \theta \\0&0&\sin \theta &\cos \theta \end{pmatrix}},}$$symmetric and orthogonal, respectively.

Weyl spinors and bispinors

Solutions to the Dirac equation transform under the (⁠1/2⁠, 0) ⊕ (0, ⁠1/2⁠)-representation. Dirac discovered the gamma matrices in his search for a relativistically invariant equation, then already known to mathematicians.^[109]

By taking, in turn, m = ⁠1/2⁠, n = 0 and m = 0, n = ⁠1/2⁠ and by setting $${\displaystyle J_{i}^{\left({\frac {1}{2}}\right)}={\frac {1}{2}}\sigma _{i}}$$in the general expression (G1), and by using the trivial relations 1₁ = 1 and J⁽⁰⁾ = 0, it follows

These are the left-handed and right-handed Weyl spinor representations. They act by matrix multiplication on 2-dimensional complex vector spaces (with a choice of basis) V_L and V_R, whose elements Ψ_L and Ψ_R are called left- and right-handed Weyl spinors respectively. Given $${\displaystyle \left(\pi _{\left({\frac {1}{2}},0\right)},V_{\text{L}}\right)\quad {\text{and}}\quad \left(\pi _{\left(0,{\frac {1}{2}}\right)},V_{\text{R}}\right)}$$their direct sum as representations is formed,^[160]

This is, up to a similarity transformation, the (⁠1/2⁠,0) ⊕ (0,⁠1/2⁠) Dirac spinor representation of ${\displaystyle {\mathfrak {so}}(3;1).}$ It acts on the 4-component elements (Ψ_L, Ψ_R) of (V_L ⊕ V_R), called bispinors, by matrix multiplication. The representation may be obtained in a more general and basis independent way using Clifford algebras. These expressions for bispinors and Weyl spinors all extend by linearity of Lie algebras and representations to all of ${\displaystyle {\mathfrak {so}}(3;1).}$ Expressions for the group representations are obtained by exponentiation.

Open problems

The classification and characterization of the representation theory of the Lorentz group was completed in 1947. But in association with the Bargmann–Wigner programme, there are yet unresolved purely mathematical problems, linked to the infinite-dimensional unitary representations.

The irreducible infinite-dimensional unitary representations may have indirect relevance to physical reality in speculative modern theories since the (generalized) Lorentz group appears as the little group of the Poincaré group of spacelike vectors in higher spacetime dimension. The corresponding infinite-dimensional unitary representations of the (generalized) Poincaré group are the so-called tachyonic representations. Tachyons appear in the spectrum of bosonic strings and are associated with instability of the vacuum.^[161][162] Even though tachyons may not be realized in nature, these representations must be mathematically understood in order to understand string theory. This is so since tachyon states turn out to appear in superstring theories too in attempts to create realistic models.^[163]

One open problem is the completion of the Bargmann–Wigner programme for the isometry group SO(D − 2, 1) of the de Sitter spacetime dS_D−2. Ideally, the physical components of wave functions would be realized on the hyperboloid dS_D−2 of radius μ > 0 embedded in ${\displaystyle \mathbb {R} ^{D-2,1}}$ and the corresponding O(D−2, 1) covariant wave equations of the infinite-dimensional unitary representation to be known.^[162]

See also

• Bargmann–Wigner equations
• Dirac algebra
• Gamma matrices
• Lorentz group
• Möbius transformation
• Poincaré group
• Representation theory of the Poincaré group
• Symmetry in quantum mechanics
• Wigner's classification

Remarks

1. The way in which one represents the spacetime symmetries may take many shapes depending on the theory at hand. While not being the present topic, some details will be provided in footnotes labeled "nb", and in the section applications.
2. Weinberg 2002, p. 1 "If it turned out that a system could not be described by a quantum field theory, it would be a sensation; if it turned out it did not obey the rules of quantum mechanics and relativity, it would be a cataclysm."
3. In 1945 Harish-Chandra came to see Dirac in Cambridge. Harish-Chandra became convinced that theoretical physics was not the field he should be in. He had found an error in a proof by Dirac in his work on the Lorentz group. Dirac said "I am not interested in proofs but only interested in what nature does." Harish-Chandra later wrote "This remark confirmed my growing conviction that I did not have the mysterious sixth sense which one needs in order to succeed in physics and I soon decided to move over to mathematics." Dirac did however suggest the topic of Harish-Chandra's thesis, the classification of the irreducible infinite-dimensional representations of the Lorentz group. See Dalitz & Peierls 1986
4. See formula (1) in S-matrix#From free particle states for how free multi-particle states transform.
5. Weinberg 2002, Equations 5.1.4–5. Weinberg deduces the necessity of creation and annihilation operators from another consideration, the cluster decomposition principle, Weinberg (2002, Chapter 4.)
6. A prescription for how the particle should behave under CPT symmetry may be required as well.
7. For instance, there are versions (free field equations, i.e. without interaction terms) of the Klein–Gordon equation, the Dirac equation, the Maxwell equations, the Proca equation, the Rarita–Schwinger equation, and the Einstein field equations that can systematically be deduced by starting from a given representation of the Lorentz group. In general, these are collectively the quantum field theory versions of the Bargmann–Wigner equations.

   See Weinberg (2002, Chapter 5), Tung (1985, Section 10.5.2) and references given in these works.

   It should be remarked that high spin theories (s > 1) encounter difficulties. See Weinberg (2002, Section 5.8), on general (m, n) fields, where this is discussed in some depth, and references therein. High spin particles do without a doubt exist, e.g. nuclei, the known ones are just not elementary.

8. For part of their representation theory, see Bekaert & Boulanger (2006), which is dedicated to representation theory of the Poincare group. These representations are obtained by the method of induced representations or, in physics parlance, the method of the little group, pioneered by Wigner in 1939 for this type of group and put on firm mathematical footing by George Mackey in the fifties.
9. Hall (2015, Section 4.4.)

   One says that a group has the complete reducibility property if every representation decomposes as a direct sum of irreducible representations.

10. Dirac suggested the topic of Wigner (1939) as early as 1928 (as acknowledged in Wigner's paper). He also published one of the first papers on explicit infinite-dimensional unitary representations in Dirac (1945) (Langlands 1985), and suggested the topic for Harish-Chandra's thesis classifying irreducible infinite-dimensional representations (Dalitz & Peierls 1986).
11. Knapp 2001 The rather mysterious looking third isomorphism is proved in chapter 2, paragraph 4.
12. Tensor products of representations, π_g ⊗ π_h of ${\displaystyle {\mathfrak {g}}\oplus {\mathfrak {h}}}$ can, when both factors come from the same Lie algebra ${\displaystyle {\mathfrak {h}}={\mathfrak {g}},}$ either be thought of as a representation of ${\displaystyle {\mathfrak {g}}}$ or ${\displaystyle {\mathfrak {g}}\oplus {\mathfrak {g}}}$.
13. When complexifying a complex Lie algebra, it should be thought of as a real Lie algebra of real dimension twice its complex dimension. Likewise, a real form may actually also be complex as is the case here.
14. Combine Weinberg (2002, Equations 5.6.7–8, 5.6.14–15) with Hall (2015, Proposition 4.18) about Lie algebra representations of group tensor product representations.
15. The "traceless" property can be expressed as S_αβg^αβ = 0, or S_α^α = 0, or S^αβg_αβ = 0 depending on the presentation of the field: covariant, mixed, and contravariant respectively.
16. This doesn't necessarily come symmetric directly from the Lagrangian by using Noether's theorem, but it can be symmetrized as the Belinfante–Rosenfeld stress–energy tensor.
17. This is provided parity is a symmetry. Else there would be two flavors, (⁠3/2⁠, 0) and (0, ⁠3/2⁠) in analogy with neutrinos.
18. The terminology differs between mathematics and physics. In the linked article term projective representation has a slightly different meaning than in physics, where a projective representation is thought of as a local section (a local inverse) of the covering map from the covering group onto the group being covered, composed with a proper representation of the covering group. Since this can be done (locally) continuously in two ways in the case at hand as explained below, the terminology of a double-valued or two-valued representation is natural.
19. In particular, A commutes with the Pauli matrices, hence with all of SU(2) making Schur's lemma applicable.
20. Meaning the kernel is trivial, to see this recall that the kernel of a Lie algebra homomorphism is an ideal and hence a subspace. Since p is 2:1 and both ${\displaystyle {\text{SL}}(2,\mathbb {C} )}$ and SO(3; 1)⁺ are 6-dimensional, the kernel must be 0-dimensional, hence {0}.
21. The exponential map is one-to-one in a neighborhood of the identity in ${\displaystyle {\text{SL}}(2,\mathbb {C} ),}$ hence the composition ${\displaystyle \exp \circ \sigma \circ \log :{\text{SL}}(2,\mathbb {C} )\to {\text{SO}}(3;1)^{+},}$ where σ is the Lie algebra isomorphism, is onto an open neighborhood U ⊂ SO(3; 1)⁺ containing the identity. Such a neighborhood generates the connected component.
22. Rossmann 2002 From Example 4 in section 2.1 : This can be seen as follows. The matrix q has eigenvalues {−1, −1}, but it is not diagonalizable. If q = exp(Q), then Q has eigenvalues λ, −λ with λ = iπ + 2πik for some k because elements of ${\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )}$ are traceless. But then Q is diagonalizable, hence q is diagonalizable, which is a contradiction.
23. Rossmann 2002, Proposition 10, paragraph 6.3. This is easiest proved using character theory.
24. Any discrete normal subgroup of a path connected group G is contained in the center Z of G.

    Hall 2015, Exercise 11, chapter 1.

25. A semisimple Lie group does not have any non-discrete normal abelian subgroups. This can be taken as the definition of semisimplicity.
26. A simple group does not have any non-discrete normal subgroups.
27. By contrast, there is a trick, also called Weyl's unitarian trick, but unrelated to the unitarian trick of above showing that all finite-dimensional representations are, or can be made, unitary. If (Π, V) is a finite-dimensional representation of a compact Lie group G and if (·, ·) is any inner product on V, define a new inner product (·, ·)_Π by (x, y)_Π = ∫_G(Π(g)x, Π(g)y dμ(g), where μ is Haar measure on G. Then Π is unitary with respect to (·, ·)_Π. See Hall (2015, Theorem 4.28.)

    Another consequence is that every compact Lie group has the complete reducibility property, meaning that all its finite-dimensional representations decompose as a direct sum of irreducible representations. Hall (2015, Definition 4.24., Theorem 4.28.)

    It is also true that there are no infinite-dimensional irreducible unitary representations of compact Lie groups, stated, but not proved in Greiner & Müller (1994, Section 15.2.).

28. Lee 2003 Lemma A.17 (c). Closed subsets of compact sets are compact.
29. Lee 2003 Lemma A.17 (a). If f : X → Y is continuous, X is compact, then f(X) is compact.
30. The non-unitarity is a vital ingredient in the proof of the Coleman–Mandula theorem, which has the implication that, contrary to in non-relativistic theories, there can exist no ordinary symmetry relating particles of different spin. See Weinberg (2000)
31. This is one of the conclusions of Cartan's theorem, the theorem of the highest weight.
    Hall (2015, Theorems 9.4–5.)
32. Hall 2015, Section 8.2 The root system is the union of two copies of A₁, where each copy resides in its own dimensions in the embedding vector space.
33. Rossmann 2002 This definition is equivalent to the definition in terms of the connected Lie group whose Lie algebra is the Lie algebra of the root system under consideration.
34. See Simmons (1972, Section 30.) for precise conditions under which two Frobenius method yields two linearly independent solutions. If the exponents do not differ by an integer, this is always the case.
35. "This is as close as one comes to the source of the theory of infinite-dimensional representations of semisimple and reductive groups...", Langlands (1985, p. 204.), referring to an introductory passage in Dirac's 1945 paper.
36. Note that for a Hilbert space H, HS(H) may be identified canonically with the Hilbert space tensor product of H and its conjugate space.
37. If finite-dimensionality is demanded, the results is the (m, n) representations, see Tung (1985, Problem 10.8.) If neither is demanded, then a broader classification of all irreducible representations is obtained, including the finite-dimensional and the unitary ones. This approach is taken in Harish-Chandra (1947).

Notes

1. Bargmann & Wigner 1948
2. Bekaert & Boulanger 2006
3. Misner, Thorne & Wheeler 1973
4. Weinberg 2002, Section 2.5, Chapter 5.
5. Tung 1985, Sections 10.3, 10.5.
6. Tung 1985, Section 10.4.
7. Dirac 1945
8. Harish-Chandra 1947
9. Greiner & Reinhardt 1996, Chapter 2.
10. Weinberg 2002, Foreword and introduction to chapter 7.
11. Weinberg 2002, Introduction to chapter 7.
12. Tung 1985, Definition 10.11.
13. Greiner & Müller (1994, Chapter 1)
14. Greiner & Müller (1994, Chapter 2)
15. Tung 1985, p. 203.
16. Delbourgo, Salam & Strathdee 1967
17. Weinberg (2002, Section 3.3)
18. Weinberg (2002, Section 7.4.)
19. Tung 1985, Introduction to chapter 10.
20. Tung 1985, Definition 10.12.
21. Tung 1985, Equation 10.5-2.
22. Weinberg 2002, Equations 5.1.6–7.
23. Tung 1985, Equation 10.5–18.
24. Weinberg 2002, Equations 5.1.11–12.
25. Tung 1985, Section 10.5.3.
26. Zwiebach 2004, Section 6.4.
27. Zwiebach 2004, Chapter 7.
28. Zwiebach 2004, Section 12.5.
29. Weinberg 2000, Section 25.2.
30. Zwiebach 2004, Last paragraph, section 12.6.
31. These facts can be found in most introductory mathematics and physics texts. See e.g. Rossmann (2002), Hall (2015) and Tung (1985).
32. Hall (2015, Theorem 4.34 and following discussion.)
33. Wigner 1939
34. Hall 2015, Appendix D2.
35. Greiner & Reinhardt 1996
36. Weinberg 2002, Section 2.6 and Chapter 5.
37. Coleman 1989, p. 30.
38. Lie 1888, 1890, 1893. Primary source.
39. Coleman 1989, p. 34.
40. Killing 1888 Primary source.
41. Rossmann 2002, Historical tidbits scattered across the text.
42. Cartan 1913 Primary source.
43. Green 1998, p=76.
44. Brauer & Weyl 1935 Primary source.
45. Tung 1985, Introduction.
46. Weyl 1931 Primary source.
47. Weyl 1939 Primary source.
48. Langlands 1985, pp. 203–205
49. Harish-Chandra 1947 Primary source.
50. Tung 1985, Introduction
51. Wigner 1939 Primary source.
52. Klauder 1999
53. Bargmann 1947 Primary source.
54. Bargmann was also a mathematician. He worked as Albert Einsteins assistant at the Institute for Advanced Study in Princeton (Klauder (1999)).
55. Bargmann & Wigner 1948 Primary source.
56. Dalitz & Peierls 1986
57. Dirac 1928 Primary source.
58. Weinberg 2002, Equations 5.6.7–8.
59. Weinberg 2002, Equations 5.6.9–11.
60. Hall 2003, Chapter 6.
61. Knapp 2001
62. This is an application of Rossmann 2002, Section 6.3, Proposition 10.
63. Knapp 2001, p. 32.
64. Weinberg 2002, Equations 5.6.16–17.
65. Weinberg 2002, Section 5.6. The equations follow from equations 5.6.7–8 and 5.6.14–15.
66. Tung 1985
67. Lie 1888
68. Rossmann 2002, Section 2.5.
69. Hall 2015, Theorem 2.10.
70. Bourbaki 1998, p. 424.
71. Weinberg 2002, Section 2.7 p.88.
72. Weinberg 2002, Section 2.7.
73. Hall 2015, Appendix C.3.
74. Wigner 1939, p. 27.
75. Gelfand, Minlos & Shapiro 1963 This construction of the covering group is treated in paragraph 4, section 1, chapter 1 in Part II.
76. Rossmann 2002, Section 2.1.
77. Hall 2015, First displayed equations in section 4.6.
78. Hall 2015, Example 4.10.
79. Knapp 2001, Chapter 2.
80. Knapp 2001 Equation 2.1.
81. Hall 2015, Equation 4.2.
82. Hall 2015, Equation before 4.5.
83. Knapp 2001 Equation 2.4.
84. Knapp 2001, Section 2.3.
85. Hall 2015, Theorems 9.4–5.
86. Weinberg 2002, Chapter 5.
87. Hall 2015, Theorem 10.18.
88. Hall 2003, p. 235.
89. See any text on basic group theory.
90. Rossmann 2002 Propositions 3 and 6 paragraph 2.5.
91. Hall 2003 See exercise 1, Chapter 6.
92. Bekaert & Boulanger 2006 p.4.
93. Hall 2003 Proposition 1.20.
94. Lee 2003, Theorem 8.30.
95. Weinberg 2002, Section 5.6, p. 231.
96. Weinberg 2002, Section 5.6.
97. Weinberg 2002, p. 231.
98. Weinberg 2002, Sections 2.5, 5.7.
99. Tung 1985, Section 10.5.
100. Weinberg 2002 This is outlined (very briefly) on page 232, hardly more than a footnote.
101. Hall 2003, Proposition 7.39.
102. Hall 2003, Theorem 7.40.
103. Hall 2003, Section 6.6.
104. Hall 2003, Second item in proposition 4.5.
105. Hall 2003, p. 219.
106. Rossmann 2002, Exercise 3 in paragraph 6.5.
107. Hall 2003 See appendix D.3
108. Weinberg 2002, Equation 5.4.8.
109. Weinberg 2002, Section 5.4.
110. Weinberg 2002, pp. 215–216.
111. Weinberg 2002, Equation 5.4.6.
112. Weinberg 2002 Section 5.4.
113. Weinberg 2002, Section 5.7, pp. 232–233.
114. Weinberg 2002, Section 5.7, p. 233.
115. Weinberg 2002 Equation 2.6.5.
116. Weinberg 2002 Equation following 2.6.6.
117. Weinberg 2002, Section 2.6.
118. For a detailed discussion of the spin 0, ⁠1/2⁠ and 1 cases, see Greiner & Reinhardt 1996.
119. Weinberg 2002, Chapter 3.
120. Rossmann 2002 See section 6.1 for more examples, both finite-dimensional and infinite-dimensional.
121. Gelfand, Minlos & Shapiro 1963
122. Churchill & Brown 2014, Chapter 8 pp. 307–310.
123. Gonzalez, P. A.; Vasquez, Y. (2014). "Dirac Quasinormal Modes of New Type Black Holes in New Massive Gravity". Eur. Phys. J. C. 74:2969 (7): 3. arXiv:1404.5371. Bibcode:2014EPJC...74.2969G. doi:10.1140/epjc/s10052-014-2969-1. ISSN 1434-6044. S2CID 118725565.
124. Abramowitz & Stegun 1965, Equation 15.6.5.
125. Simmons 1972, Sections 30, 31.
126. Simmons 1972, Sections 30.
127. Simmons 1972, Section 31.
128. Simmons 1972, Equation 11 in appendix E, chapter 5.
129. Langlands 1985, p. 205.
130. Varadarajan 1989, Sections 3.1. 4.1.
131. Langlands 1985, p. 203.
132. Varadarajan 1989, Section 4.1.
133. Gelfand, Graev & Pyatetskii-Shapiro 1969
134. Knapp 2001, Chapter II.
135. Taylor 1986
136. Knapp 2001 Chapter 2. Equation 2.12.
137. Bargmann 1947
138. Gelfand & Graev 1953
139. Gelfand & Naimark 1947
140. Takahashi 1963, p. 343.
141. Knapp 2001, Equation 2.24.
142. Folland 2015, Section 3.1.
143. Folland 2015, Theorem 5.2.
144. Tung 1985, Section 10.3.3.
145. Harish-Chandra 1947, Footnote p. 374.
146. Tung 1985, Equations 7.3–13, 7.3–14.
147. Harish-Chandra 1947, Equation 8.
148. Hall 2015, Proposition C.7.
149. Hall 2015, Appendix C.2.
150. Tung 1985, Step II section 10.2.
151. Tung 1985, Equations 10.3–5. Tung's notation for Clebsch–Gordan coefficients differ from the one used here.
152. Tung 1985, Equation VII-3.
153. Tung 1985, Equations 10.3–5, 7, 8.
154. Tung 1985, Equation VII-9.
155. Tung 1985, Equations VII-10, 11.
156. Tung 1985, Equations VII-12.
157. Tung 1985, Equations VII-13.
158. Weinberg 2002, Equation 2.4.12.
159. Weinberg 2002, Equations 2.4.18–2.4.20.
160. Weinberg 2002, Equations 5.4.19, 5.4.20.
161. Zwiebach 2004, Section 12.8.
162. Bekaert & Boulanger 2006, p. 48.
163. Zwiebach 2004, Section 18.8.

Freely available online references

• Bekaert, X.; Boulanger, N. (2006). "The unitary representations of the Poincare group in any spacetime dimension". arXiv:hep-th/0611263. Expanded version of the lectures presented at the second Modave summer school in mathematical physics (Belgium, August 2006).
• Curtright, T L; Fairlie, D B; Zachos, C K (2014), "A compact formula for rotations as spin matrix polynomials", SIGMA, 10: 084, arXiv:1402.3541, Bibcode:2014SIGMA..10..084C, doi:10.3842/SIGMA.2014.084, S2CID 18776942 Group elements of SU(2) are expressed in closed form as finite polynomials of the Lie algebra generators, for all definite spin representations of the rotation group.

References

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